Convex Analysis in General Vector Spaces

Introduction

xvn

tion, cp = m a x { / i , . . . , / „ } and tp = A 0 / 2 - Besides classical conditions one points out very recent ones. For the proof one constructs an adequate perturbation function and uses the fundamental duality theorem. In Section 2.9 we apply the fundamental duality theorem for obtaining necessary and sufficient optimality conditions in convex optimization problems with constraints. These conditions involve the subdifferentials of the functions considered or the corresponding Lagrangian. The results are well-known. However we mention the formula for the normal cone to a level set stated in Corollary 2.9.5 for not necessarily finite-valued functions which is quite new. The minimax theorem presented in Section 2.10 will be used in the section dedicated to monotone multifunctions. Throughout Chapter 3 the involved spaces are normed spaces. In Section 3.1 besides the classical theorems of Borwein, Br0ndsted-Rockafellar, Bishop-Phelps and Rockafellar (on the maximal monotonicity of the subdifferential of a convex function) we present a recent theorem of Simons and use it for a very short proof of Rockafellar's theorem (mentioned before). As a consequence of the Br0ndsted-Rockafellar theorem we obtain other three conditions for the validity of the formulas for the conjugate and subdifferential of the function F(-,A(-)) (and therefore for the functions

f + go A and f + g). The aim of Section 3.2 is to characterize the convex functions using other types of subdifferentials. In fact we use an abstract subdifferential. An example of such subdifferential is Clarke's one for which we establish several properties. The main tool for such characterizations is the well-known Zagrodny's approximate mean value theorem; the version we present subsumes several results met in the literature. We present also an integration theorem of Thibault and Zagrodny which yields the fact that two lower semicontinuous convex functions on a Banach space which has the same Fenchel subdifferential coincide up to an additive constant. In Section 3.3 we introduce the class A of functions tp : K+ —> R+ with y>(0) = 0 and several useful subclasses. To any ip € A we associate tp# € A defined by 0}. These classes of functions turn out to be useful in studying well-conditioned convex functions, uniformly convex and uniformly smooth convex functions, as well as in the study of the geometry of normed spaces. As an illustration of the use of these classes of functions we study the differentiability of convex functions with

XV111

Introduction

respect to arbitrary bornologies. Using one of the characterizations and the Br0ndsted-Rockafellar theorem one obtains the following interesting result of Asplund and Rockafellar: Let X be a Banach space and / : I - > l a proper lower semicontinuous convex function; if / * is Frechet differentiable at x* e int(dom/*) then V/*(x*) € X. In Section 3.4 we introduce the well-conditioned convex functions and give several characterizations of this notion using the conjugate and the subdifferential of the function. An important special case is that of wellconditioning with linear rate. This situation is studied in Section 3.10. In Section 3.5 we study uniformly convex and uniformly smooth convex functions, respectively. To any convex function / one associates the gages pj and 07 of uniform convexity and uniform smoothness, respectively. The gage pf has an important property: the mapping 0 < t i-» t~2pf(t) is nondecreasing. Because a/* = ( p / ) * and a/ = (p/*) # f° r a n v proper lower semicontinuous convex function / , the mapping 0 < i 4 t~2crf(t) is nonincreasing for such a function; moreover, one obtains that for such an / , / is uniformly convex if and only if / * is uniformly smooth and / is uniformly smooth if and only if / * is uniformly convex. Then one establishes many characterizations of uniformly convex functions and of uniformly smooth convex functions. In these characterizations appear functions (gages or moduli) belonging to different subclasses of A introduced in Section 3.3. These gages and moduli are sharp enough in order to obtain that / is c-strongly convex if and only if /* is Frechet differentiable on X* and V / * is c _1 -Lipschitz. Even if the results are established in general Banach spaces the natural framework for uniformly convex and uniformly smooth convex function is that of reflexive Banach spaces. This is due to the fact that when there exists a proper lower semicontinuous and uniformly convex function on a Banach space whose domain has nonempty interior, the space is necessarily reflexive. Section 3.6 is dedicated to the study of those convex functions which are uniformly convex on bounded sets and uniformly smooth on bounded sets, respectively. Under strong coercivity of the function one shows that these notions are dual. In Section 3.7 we study the function / v : X ->• E, f^x) = / „ " tp(t) dt, where

Introduction

xix

One obtains also characterizations of (local) uniform convexity and (local) uniform smoothness of X with the help of the properties of fv. For example one obtains: X is uniformly convex •£> X* is uniformly smooth •& fv is uniformly convex on bounded sets -O (fv)* is uniformly smooth on bounded sets •& {ftp)* is Frechet differentiable and V ( / v ) * is uniformly continuous on bounded sets. Note that a part of the results of this section can be found in the book [Cioranescu (1990)], but the proofs are different; note also that some notions are introduced differently in Cioranescu's book. Another application of convex analysis is emphasized in Section 3.8; here we apply the results on the existence, the uniqueness and the characterizations of optimal solutions of convex programs to the problem of the best approximation with elements of a convex subset of a normed space. In Section 3.9 it is shown that there exists a strong relationship between the well-posedness of the minimization problem min/(:r) s.t. x £ X, and the differentiability at 0 of the conjugate / * of / ; when / is convex these properties are equivalent. Using this result we establish a very interesting characterization of Chebyshev sets in Hilbert spaces and show that the class of weakly closed Chebyshev sets coincides with the class of closed convex sets in Hilbert spaces. Section 3.10 deals with sets of weak sharp minima, well-behaved convex functions and the study of the existence of global error bounds for convex inequalities. These notions were studied separately for a time, but they are intimately related. As noted above, argmin / is a set of weak sharp minima for / exactly when / is well-conditioned with linear rate. But the inequality f{x) < 0 has a global error bound exactly when argmin[/] + is a set of weak sharp minima for [/] + := max(/, 0). We give several characterizations of the fact that argmin / is a set of weak sharp minima for / , one of them being the fact that up to a constant, the conjugate / * is sublinear on a neighborhood of the origin. Several numbers associated to a convex function are introduced which are related to the conditioning number from numerical analysis. Although the most part of the results from this section are stated in the literature in finite dimensional spaces, we present them in infinite dimensions. The last section of this book, Section 3.11, is dedicated to the study of monotone multifunctions on Banach spaces. We use in the presentation two recent articles of Simons. The proofs are quite technical and use the lower semicontinuous convex function XM associated to the multifunction M :

XX

Introduction

X =} X*, the minimax theorem and a few results of convex analysis. One obtains: two characterizations of maximal monotone multifunctions; the fact that the condition 0 6 int(domTi — domT2) is equivalent to other three conditions involving dom Tj and dom XT{ , and is sufficient for the maximal monotonicity of T\ + T2; dom T and Im T are convex if X is reflexive and T is maximal monotone; domT is convex if int(domT) ^ 0 and T is maximal monotone; T is locally bounded at XQ € (co(domX'))1 if T is a monotone multifunction; Rockafellar's theorem on the local boundedness of maximal monotone multifunctions. The result stating that for a maximal monotone multifunction T on the Banach space X for which dom T is convex the local boundedness of T at x 6 dom T implies that x € int (dom T) seems to be new. When applied to the subdifferential of a proper lower semicontinuous convex function / on the Banach space X, this result gives (for example): / is continuous &• d o m / is open <& df is locally bounded at any x £ d o m / . The exercises are intended to exemplify the topics treated in the book. Many exercises are auxiliary results spread in recent articles, although some of them are extracted from other books. Some exercises are important results which could be parts of textbooks, but which do not fit very well with the aim of the present book. Among them we mention Exercise 3.11 on the Moreau regularization.

Chapter 1

Preliminary Results on Functional Analysis

1.1

Preliminary Notions and Results

In this section we introduce several notions and results on separation of sets as well as some properties of topological vector spaces and locally convex spaces which are frequently used throughout the book, for easy reference. Let X be a real linear (vector) space. Throughout this work we shall use the following notation (x, y being elements of X): [x, y] := {(1 — A)x + Xy \ A e [0,1]}, [x,y[:= {(1 - X)x + \y | A € [0,1[}, ]x,y[:= {(1 - X)x + Xy | A 6]0,1[}, called closed, semi-closed and open segment, respectively. Note that [x,x] =]x,x[= {x}\ If 0 ^ A, B C X, the Minkowski sum of A and B is A + B := {a + b | a £ A,b £ B). Moreover, if x £ X, X £ R and 0 ^ T C R, then x + A := A + x := A + {x}, A • A = {70 | A € A, a £ A} and XA — {A} • A. We shall consider that A + 0 = 0 and A • 0 = 0 • A = 0. A nonempty set A C X is star-shaped at a (G A) if [a, x] C A for all 2: £ A; A is convex if [x, y] C A for all x,y £ A; Ais a, cone if 1+ • A C A (in particular 0 € A when A is a cone), R+ is the set of nonnegative reals; A is afflne if Ax + (1 - X)y 6 A for all x,y e A and A e R; A is balanced if Ax € ^4 for all x e A and A £ [-1,1]; A is symmetric if A = - A . Hence ^4 is balanced if and only if A is symmetric and star-shaped at 0. We consider that the empty set is convex and afflne. It is easy to prove that A is afflne o 3 a 6 l , 3XQ linear subspace of X : A = a + X0 <=>Va€^4(3a€^4) : A — a is a linear subspace. When A is affine and a £ A, the linear space XQ := A — a is called the l

2

Preliminary

results on functional

analysis

linear space parallel to A; we consider that the dimension of A is dimXoNote that if (Aj)j € / is a family of affine, convex, balanced subsets or cones of X then f]ieIAi has the same property (Exercise!); we use the usual convention that HieO7^ = -^- Taking into account this remark, we can introduce the notions of affine, convex and conic hull of a set. So, the affine, convex and conic hull of the subset A of X are: aff A := [){V C X \ A C V, V affine}, co A := P){C C X | A C C, C convex}, cone A := f]{C CX\AcC,

C cone},

respectively. Of course, the linear hull of the subset A of X is the linear subspace spanned by A: linA := P | { ^ o C X \ A C X0, X0 linear subspace of X}. It is easy to verify (Exercise!) that aff A = { V "

XiXi n G N, (Ai)i< i < n C R, V "

X, = 1, (Xi)l
coA= {Yl^-^i

n

S N, (Aj) C K+, (xt) C A, Yl"-^

C A> , = 1

) '

cone A - {Ax | A > 0, x £ A} = 1+ • A, where N is the set of positive integers. Let us mention some properties of the affine and convex hulls. Consider Y another linear space, T : X -> Y a linear operator, A,B c X, C C Y nonempty sets. Then: (i) aff(AxC) = aff Ax aff C; (ii) aff T(A) = T(aff A) (iii) aft(A + B) = aff A + aff B; (iv) Vo € A : aff A = a + aff (A - A) (v) aff (A - A) = UA>oA(A - A) if A is convex; (vi) aff A = linA if 0 e A (vii) co(A x C) = co A x coC; (viii) co T(A) = T(coA); (ix) co(A + 5 ) = co A + coB; (x) co(cone A) = cone(co A) (Exercises!). Let M C X be a linear subspace, and let A C X be nonempty; the algebraic interior of A with respect to M is a i n t M ^ : = {a 6 X | Vrr € M, 36 > 0, VA € [0,6] : a + Xxe It is clear that aint^f Ac

A}.

A and that M C aff (A - A) when aintM i ^ D -

Preliminary

notions and results

3

We distinguish two important cases: (i) M = X; in this case we write A instead of aintM A; A1 is called the algebraic interior of A, (ii) M = aS(A — A); in this case aintM A is denoted by M and is called the relative algebraic interior of A. Therefore a £ A1 ii and only if aff A = X and a G 14 (Exercise!). When the set A is convex we have (Exercise!) that: %

V a G i : \m(A-a)

=

cone(A-A),

whence aff A = a + cone(A — A) for every a £ A (hence cone(A — A) is the linear subspace parallel to aff A), a e i ' ^ V i e l , 3 A > 0 : a + Xx £ A& cone(A - a) = X, and a G M o V a ; G A, 3 A > 0 : (1 + X)a - Xx G A <$ cone(yl — a) = cone(A — A) <=> cone(A — a) is a linear subspace O M

n(C — a) is a linear subspace;

(1.1)

the dimension of the convex set A is dim A := dim(aff A) = dim (cone(A — A)). Some properties of the algebraic interior are listed below. Let 0 ^ A,B CX, i s l a n d A G E \ {0}; then: (i) *(x + A) =x + iA; (ii) ^XA) = X • % (iii) A + Bi C (A + B)'; (iv) A + Bl = {A + BY if Bi = B; (v) iA + iB c 1{A + B); (vi) *(A. + B) = A + *B if A, B are convex, VI / 0 and *B T^ 0; (vii) M 7^ 0 if A is convex and dim A < 00; (viii) if A is convex then [a, x[ C VI for all a G M. and i £ A In the sequel the results will be established for real topological vector spaces (tvs for short) or real locally convex spaces (lcs for short). When X is a tvs it is well-known that the class Nx of closed and balanced neighborhoods of 0 G X is a base of neighborhoods of 0; when X is a lcs then the class J^cx of the closed, convex and balanced neighborhoods of 0 G X is also a base of neighborhoods of 0. If X, Y are real linear spaces, we denote by L(X, Y) the real linear space of linear operators from X into Y. The space L(X,R) is denoted by X' and is called the algebraic dual of X; an element of X' is called a linear functional. When X, Y are topological vector spaces, we denote by &{X, Y)

4

Preliminary

results on functional

analysis

the linear space of continuous linear operators from X into Y; the space £ ( X , E) is denoted by X* and is called the topological dual of X. Let now A be an absorbing subset of the linear space X, i.e. 0 £ A1; the Minkowski gauge of A is defined by pA : X -> M,

^ ( a : ) := inf{A > 0 | a; £

\A).

It is obvious that PA = P[o,i].4- Moreover, if A, B C X and C C Y are absorbing and star-shaped sets, where Y is another linear space, one has (Exercise!): {x 6 X | p A (a) < 1} C A C {x € X | P A ( Z ) < 1}, VxGX

: pAns(a;) = max {PA (a;), p B (a;)},

Va; 6 X, 2/ € Y : PAxc(x,y)

=

max{pA(x),pc(y)}-

Other useful properties of the Minkowski gauge are mentioned in the next result. Recall that p : X -» M is sublinear if p(0) = 0, p(x + y) < p(x) +p(y) [with the convention (+oo) + (—oo) = +oo] and p(Xx) = Xp(x) for all x, y £ X, X € P := ]0, oo[; p is a semi-norm if p is a finite, sublinear and even [i.e. p(—x) = p(x) for every x € X] function. Proposition 1.1.1 space X.

Let A be a convex and absorbing subset of the linear

(i) Then PA is finite, sublinear and A1 = {x E X \ PA{X) < 1}; furthermore, if A is symmetric then PA is a semi-norm, too. (ii) Assume, moreover, that X is a topological vector space and V is a neighborhood of Q £ X. Then pv is continuous and intV = {xeX\pv{x)

< 1},

clV = {xeX

\pv{x)

< 1}.

The following result will be useful, too. Theorem 1.1.2 X. Then

Let C be a convex subset of the topological vector space

(i) clC is convex; (ii)

if a £ int C and x £ cl C, then [a, x[ C int C;

(iii) intC is convex; (iv)

if int C ^ 0 then cl(int C) = cl C and int(cl C) = int C;

(v)

if int C ^ 0 then Cl = int C.

Preliminary

notions and results

5

Using the Minkowski gauge one obtains the geometrical versions of the Hahn-Banach theorem, i.e. separation theorems. In the sequel we give several separation theorems for convex subsets of topological vector spaces or locally convex space. Theorem 1.1.3 (Eidelheit) Let A and B be two nonempty convex subsets of the topological vector space X. If int A ^ 0 and B flint A = 0 then there exist x* e X* \ {0} and a £ E such that VxeA,\/yeB

: (x,x*) < a < (y,x*),

(1.2)

or equivalently, supa;*(yl) < inf x*(B). The separation condition (1.2) can be given in a different manner. Let x* e X* \ {0} and a £ t Consider the sets H^a:={x£X\(x,x*)
(x,x*)


Hx*,a := {x e X \ (x,x*) = a } ; similarly one defines H^. and H>.a. All these sets are convex and nonempty. The set Hx*^a is called a closed hyperplane, H<. a and H>* a are called open half-spaces, while H-. a and H~.a are called closed halfspaces. Hx-, a are open sets; moreover, c\H< a = H% a and {Hf,a)1 = int H~. >a = H< a (Exercises!). Theorem 1.1.3 states the existence of x* G X* \ {0} and a G K such that A c H-, a and B C H-, a; in this situation we say that Hx*tC[ separates A and B; the separation is proper when AUB <£. Hx*i<x and the separation is strict when A n Hx*^a = 0 or B n Hx*ia = 0. When x0 6 A and ffx*,a separates A and {xo} we say that ifx*,a is a supporting hyperplane of A at xo; XQ is called a support point and x* is called a support functional. Therefore x* € X* \ {0} is a support functional for A if and only if x* attains its supremum on A. Generally, Hx*
6

Preliminary

results on functional

analysis

In the case of locally convex spaces one has the following result for the separation of two sets. T h e o r e m 1.1.5 Let X be a locally convex space and A,B C X be two nonempty convex sets. If A is closed, B is compact and An B = 0, then there exist x* G X* \ {0} and ct\,a.2 G K such that Vx G A, Vy e B : (x,x*) < ax < a2 < (y,x*), or equivalently, sup x* (A) < inf x* (B). The two preceding results can be stated in a more general setting. T h e o r e m 1.1.6 Let A and B be two nonempty convex subsets of the topological vector space X such that int(A — B) ^ 0. Then 0 £ i n t ( A - £ ) < £ • 3x* G X * \ { 0 } : supa;*(yl) < inf x*{B). T h e o r e m 1.1.7 Let A and B be two nonempty convex subsets of the locally convex space X. Then Q$d{A-B)&3x*

eX* : supx*{A) < inf x*(B).

The preceding theorem shows the usefulness of having criteria for the closedness of the difference (or sum) of two convex sets. In order to give such a criterion, let A be a nonempty convex subset of the topological vector space X. The recession cone of A is defined by vecA := {u G X | Vo G A : a + uGA}. It is easy to show that rec A is a convex cone and A + rec A = A. When A is a closed convex set we have that vecA = f]t>ot(A-a)

(1.3)

for every a 6 A. In this case it is obvious that rec A is a closed convex cone which is also denoted by Aoo. It is easy to see that when X is a finite dimensional separated topological vector space and A is a closed convex nonempty subset of X, A^ = {0} if, and only if, A is bounded. This is no longer true when dimX ~ oo. Example 1.1.1 Let X := £p with p £ [l,oo] and A := {(x„)„>i G £p | \xn\ < n Vn G N}. It is obvious that A is a closed convex set which is not bounded because nen G A for every n G N, but A^ = {0}. Indeed, if

Preliminary

notions and results

7

u = (un) G Aoo then tu G A for every t > 0; so \tun\ < n for every t > 0, whence u„ = 0. Hence u = 0. The following famous theorem was obtained in [Dieudonne (1966)]. Theorem 1.1.8 (Dieudonne) Let A, B be nonempty closed convex subsets of the locally convex space X. If A or B is locally compact and A^ n J3oo is a linear subspace, then A — B is closed. In convex analysis (as well as in functional analysis) one often uses the following sets associated to a nonempty subset A of the locally convex space X: A°

{x*

A+

{x* ex* \Vxe A {x* ex* IVze A

A^

eX*\Vx€A

(x,x*)>-l}, (x,x*)>0}, (x,x*) = 0},

called the polar, the dual cone and the orthogonal space of A, respectively. One verifies easily that A° is a w*-closed convex set which contains 0, that A+ is a w*-closed convex cone, and, finally, that A1- is a u;*-closed linear subspace of X*, where w* = a(X*,X) is the weak* topology on X*. Similarly, for 0 ^ B C X* we define the polar, the dual cone and the orthogonal space; for example, the polar of B is B° := {x G X | Vx* G B : (x,x*) > - 1 } . It is obvious that B° is a closed convex set containing 0, B+ is a closed convex cone, and, finally, B1- is a closed linear subspace. One verifies easily that when A,BcX and A G P we have: (i) A° is convex and 0 G A°; (ii) A U {0} C (A°)° =: A°°; (hi) AcB => A° D B°; (iv) (AUB)° = A°nB°; (v) if 0 € AnB then (A + B)+ = (AUB)+ =A+n B+; (vi) (\A)° = {A°; (vii) A° = A+ if A is a cone, and A° = A+ = A1- if A is a linear subspace; (viii) (T(A))° = ( T * ) " 1 ^ 0 ) , if T G &(X,Y), where Y is another locally convex space. A very useful result is the bipolar's theorem. Let X be a topological vector space and A C X; the set coA := cl(coA) is called the closed convex hull of the set A; it is the smallest closed convex set containing A. Similarly, coneA :— cl(coneyl) is called the closed conic hull of A. Theorem 1.1.9

(bipolar) Let A be a nonempty subset of the locally con-

Preliminary

8

vex space X.

results on functional

analysis

Then

A00 = co(A U {0}),

A + + = cone(co A),

A±JL

= cl(lin A).

It follows that for the nonempty subset A of the lcs X one has: (a) A°° = A o A is closed, convex and 0 € A; (b) A + + = A <£> A is a closed convex cone; (c) A-"-1- = A <=> A is a closed linear subspace. Another famous result is the following. Theorem 1.1.10 (Alaoglu-Bourbaki) Let X be a locally convex space and U C X be a neighborhood of the origin. Then U° is w*-compact. We finish this preliminary section with some notions and results concerning completeness and metrizability of topological vector spaces. The subset A of the topological vector space X is complete (quasicomplete) if every (bounded) Cauchy net (xi)i^i C A is convergent to an element x £ A. Of course, any complete set is closed and any closed subset of a complete set is complete (Exercise!). Recall that the topological space (X, T) is first countable if every element of X has a (at most) countable base of neighborhoods. Note that a subset A of a first countable tvs X is complete if and only if every Cauchy sequence of A is convergent to an element of A; in particular, a first countable tvs is complete if and only if it is quasi-complete (Exercise!). We shall use several times the hypothesis that a certain topological vector space is first countable. The next result refers to the first countability of locally convex spaces. Proposition 1.1.11

Let (X,T) be a locally convex space. Then

(i) (X, T) is first countable O 3 7 a (at most) countable family of seminorms on X such that r = rg> O r is semi-metrizable, i.e. there exists a semi-metric d on X such that T = T^; the semi-metric d may be chosen to be invariant to translations (i.e. d(x + z,y + z) = d(x,y) for allx,y,z € X). (ii) (X, T) is separated and first countable if and only if T is metrizable, i.e. there exists a metric d on X such that r = T&. Note that when the topology r of a locally convex space X coincides with the topology T<J determined by a semi-metric d invariant to translations, the completeness with respect to r and d are equivalent. One says that the locally convex space X is a Frechet space if X is complete and metrizable. It is obvious that every closed linear subspace of a Frechet space is Frechet,

Closedness and inferiority

notions

9

too. One says that the topological vector space X is barreled if every absorbing, convex and closed subset of X is a neighborhood of 0 £ X. As application of the Baire theorem one obtains that every Frechet space is barreled. It is well-known that in a finite dimensional separated topological vector space any convex and absorbing set is a neighborhood of the origin.

1.2

Closedness and Interiority Notions

Consider X a real topological vector space. We say that the series Yln>i Xn is convergent (resp. Cauchy) if the sequence (£n)n£N is convergent (resp. Cauchy), where Sn := Y^k=i xk f° r every n £ N; of course, any convergent series is Cauchy. Let A C X; by a convex series with elements of A we mean a series of the form Ylm>i ^mXm with (Am) C IR+, {xm) C A and Y,m>i ^™ = ^ if, furthermore, the sequence (xm) is bounded we speak about a b-convex series. We say that A is cs-closed if any convergent convex series with elements of A has its sum in A;* A is cs-complete if any Cauchy convex series with elements of A is convergent and its sum is in A. Similarly, the set A is called ideally convex if any convergent b-convex series with elements of A has its sum in A and A is bcs-complete if any Cauchy b-convex series with elements of A is convergent and its sum is in A. It is obvious that any cs-closed set is ideally convex, every ideally convex set is convex, every cscomplete set is cs-closed and every complete convex set is cs-complete; if X is complete, then A C X is cs-complete (bcs-complete) if and only if A is csclosed (ideally convex). If Xo is a linear subspace of X and A is a cs-closed (ideally convex) subset of X, then Xo C\ A is a cs-closed (ideally convex) subset of Xo (endowed with the induced topology). Moreover, if A C X and B C Y are nonempty, then A x B is cs-closed (cs-complete, ideally convex, bcs-complete) if and only if A and B are cs-closed (cs-complete, ideally convex, bcs-complete). If X is first countable, a linear subspace XQ of X is cs-closed (cs-complete) if and only if it is closed (complete); moreover, if X is a locally convex space, X0 is closed if and only if Xo is ideally convex. Note also that T(A) is cs-closed (cs-complete, ideally convex, bcs-complete) if A C X is cs-closed (cs-complete, ideally convex,

'Because X may not be separated, in fact we ask that every limit of (Sn) is in A.

Preliminary

10

results on functional

analysis

bcs-complete) and T : X -> Y is an isomorphism of topological vector spaces (Exercise!), Y being another tvs. We consider that the empty set is convex, ideally convex, bcs-complete, cs-complete and cs-closed. It is worth to point out that when X is a locally convex space, every b-convex series with elements of X is Cauchy (Exercise!). The class of cs-closed sets (and consequently that of ideally convex sets) is larger than the class of closed convex sets, as the next result shows. Proposition 1.2.1

Let A C X be a nonempty convex set.

(i) / / A is closed or open then A is cs-closed. (ii) If X is separated and dim A < oo then A is cs-closed. Proof, (i) Let £ n >! A n i n be a convergent convex series with elements of A; denote by x its sum. Suppose that A is closed and fix a £ A. Then, for every n G N we have that X)fc=i ^kXk + (l — Yl'kLn+i ^*) a S A. Taking the limit for n -> oo, we obtain that x G cl A = A. Suppose now that A is open. Assume that x $. A. By Theorem 1.1.3, there exists x* G X* such that (a - x, x") > 0 for every a G A. In particular (xn — x,x*) > 0 for every n G N. Multiplying by \ n > 0 and adding for n G N we get (since A„ > 0 for some n) the contradiction

0 < ^ n > 1 A„ (xn - x, x*) = (X]„>i XnXn' x*)~ \52n>i

An

) ^' x ")

=

°'

Therefore x G A. So in both cases A is cs-closed. (ii) We prove the statement by mathematical induction on n := dim A. If n = 0 A reduces to a point; it is obvious that A is cs-closed in this case. Suppose that the statement is true if dim A < n G N U {0} and show it for dim A = n + 1. Without any loss of generality we suppose that 0 G A; then Xo := aff A is a linear subspace with dimX 0 = n + 1. Because on a finite dimensional linear space there exists a unique separated linear topology and in such spaces the interior and the algebraic interior coincide for convex sets, we have that iA — intx 0 A ^ 0. Let J2n>i ^n%n be a convergent convex series with elements of A and sum x. Assume that x fi A. Because A is convex, the set P :— {n G N | A„ > 0} is infinite, and so we may assume that P — N. Applying now Theorem 1.1.3 in XQ, there exists XQ G XQ \ {0} such that (x — x, XQ) > 0 for every x G A. But X)n>i An (xn —X,XQ) = 0. Since {xn —X,XQ) > 0 and Xn > 0 for every n, we obtain that (x n ) C AQ := A(~)Hx*t\, where A :— (X,XQ). Since

Closedness and inferiority

notions

11

dimAo < dimHx*t\ = n, from the induction hypothesis we obtain the contradiction x e A0 c A. Therefore x € A. The proof is complete. • Other properties of cs-closed and ideally convex sets are given in the following result. Proposition 1.2.2 (i) If Ai C X is cs-closed (resp. ideally convex) for every i £ I then P | i e / A{ is cs-closed (resp. ideally convex). (ii) / / Xi is a topological vector space and Ai C Xi is cs-closed (resp. ideally convex) for every i 6 I, then Ylizi -^-i is cs-closed (resp. ideally convex) in Yliei Xi (which is endowed with the product topology). Proof. The proof of (i) is immediate, while for (ii) one must take into account that a sequence (x n ) n € N C X := YlieI Xi converges t o x G X (resp. is bounded) if and only if (xln) converges to xl in Xi (resp. is bounded) for every i £ I. • We say that the subset C of Y is lower cs-closed (Ics-closed for short) if there exist a Frechet space X and a cs-closed subset B of X x Y such that C = Pry (B). Similarly, the subset C of Y is lower ideally convex (li-convex for short) if there exist a Frechet space X and an ideally convex subset B of X xY such that C = Pry (B). It is obvious that any cs-closed (resp. ideally convex) set is Ics-closed (resp. li-convex), any Ics-closed set is li-convex and any li-convex set is convex, but the converse implications are not true, generally. Note also that T(A) is Ics-closed (resp. li-convex) if A C X is Ics-closed (resp. li-convex) and T : X -> Y is an isomorphism of topological vector spaces (Exercise!). The classes of Ics-closed and li-convex sets have very good stability properties as the following results show. We give only the proofs for the "li-convex" case, that for the "Ics-closed" case being similar. Proposition 1.2.3 Suppose that Y is a Frechet space and C C F x Z is a li-convex (Ics-closed) set. Then Prz(C) is a li-convex (Ics-closed) set. Proof. By hypothesis, there exists a Frechet space X and an ideally convex subset B C X x (Y x Z) such that C = P r y x z ( B ) - Since I x 7 i s a Frechet space and Prz(C) = Prz(B), we have that Prz(C) is a li-convex subset of Z. • Proposition 1.2.4

Let I be an at most countable nonempty set.

(i) If Ci C Y is li-convex (Ics-closed) for every i £ I then (~)ieI Ci is li-convex (Ics-closed).

12

Preliminary

results on functional

analysis

(ii) / / Y{ is a topological vector space and Ci C Yi is li-convex (Icsclosed) for every i 6 I, then Yli&iCi 8S li-convex (Ics-closed) in W^jYi. Proof, (i) For each i £ I there exist X» a Frechet space and an ideally convex set Bt C X» xY such that d = PrY(Bi). The space X := fliei -%i is a Frechet space as the product of an at most countable family of Frechet spaces. Let Bi •= {{(xj)jei,y)

e X x Y | (xi,y) £ Bi} .

Then Bi is an ideally convex set by Proposition 1.2.2(h). It follows that B := f]ieI Bi is ideally convex by Proposition 1.2.2(i). Since PrY(B) = Hie/ Ci, ^ f ° u o w s that flie/ Ci is li-convex. (ii) For each i £ I there exist Xj a Frechet space and an ideally convex set Bi cXiX Yi such that d = Pry ; (Bi). By Proposition 1.2.2(h), ]JieI Bt is an ideally convex subset of r i i e / ( ^ j x ^ ) - The space X := riig/ ^i 1S a Frechet space; let Y := FJie/ ^i- Consider the set B:= {((xi)ieI,(yi)ieI)

£XxY\

(xi,yi)

£ Bi Vt 6 / } .

Since T : I l i e / ( X * x y4) - • X x y , T ( ( x i , y i ) i 6 / ) := ((xi)ieI,(yi)iei) is an isomorphism of topological vector spaces, B — T (Y\ieI Bi) is ideally convex. As C := riig/ C* = ^VY(B), C is li-convex. D Before stating other properties of li-convex and lcs-closed sets, let us define some notions and notations related to multifunctions. Let E,F be two nonempty sets; a mapping ft : E —• 2F is called a multifunction, and it will be denoted by ft : E =X F. The set domft := {x £ E \ 5l(x) -£ 0} is called the d o m a i n of the multifunction 51; the image of 51 is Imft := UigB^( a ; )i the g r a p h of 51 is the set grft :— {(x,y) \ y £ 5l(x)} C E x F; the inverse of the multifunction 51 is the multifunction ft"1 : F =} E defined by ft_1(?/) := {x £ E | y £ %(x)}. Therefore domft- 1 = Imft, Imft- 1 = domft and grft- 1 = {(y,x) | (x,y) £ grft}. Frequently we shall identify a multifunction with its graph. For A C E and B C F one defines 51(A) := \JxeA5l(x) and ft"1^) := Uj, e B# - 1 (2/); i n particular Imft = ft(.E) and domft = ft_1(F). If 8 : F =4 G is another multifunction, then the composition of the multifunctions S and 51 is the multifunction Soft :E=lG, (Soft) (a) := \Jy€Ci{x) S(y). If ft, S : E =} F and F is a linear space, the sum of ft and S is the multifunction ft + S : E =t F, (ft + S)(x) :=5i(x) + S(x).

Closedness and interiority notions

13

Let now K : X r{ 7 ; we say that 51 is convex (closed, ideally convex, bcs-complete, cs-convex, cs-complete, li-convex, lcs-closed) if its graph is a convex (closed, ideally convex, bcs-complete, cs-convex, cs-complete, li-convex, lcs-closed) subset of X x Y. Note that 31 is convex if and only if V i , a ' eX,

VAe [0,1] : \3l(x) + (l-\)3l(x')c3l(\x

Proposition 1.2.5

+

Let A, B C X, 31, S : X =4 Y and7:Y

{l-\)x'). =} Z.

(i) If X is a Frechet space and A, 31 are li-convex (resp. lcs-closed), then 01(A) is li-convex (resp. lcs-closed). (ii) If X is a Frechet space and A, B are li-convex (resp. lcs-closed), then A + B is li-convex (resp. lcs-closed). (hi) If Y is a Frechet space and 31, T are li-convex (resp. lcs-closed), then T o 31 is li-convex (resp. lcs-closed). (iv) IfY is a Frechet space and 31,$ are li-convex (resp. lcs-closed), then 31 + § is li-convex (resp. lcs-closed). Proof,

(i) We have that %{A)

=PrY((AxY)ngrJl).

Using successively Propositions 1.2.4(h), 1.2.4(i) and 1.2.3, it follows that 31(A) is li-convex. (ii) Let T : X xX —> X, T(x,y) := x + y. Since T is a continuous linear operator, grT is a closed linear subspace; in particular T is a li-convex multifunction. Since Ax B is li-convex, by (i) A + B = T(A x B) is a li-convex set. (hi) We have that gr(To3?) = P r X x 4 ( g r : R x Z) n (X x grT)). The conclusion follows from Propositions 1.2.4(h), 1.2.4(i) and 1.2.3. (iv) The sets T~{(x,z,y,y')\x€X,

y,y' eY,

z = y + y'} C (X x Y) x (Y x Y),

A:= {(x,z,y,y') \ (x,y) G g r # , y',z € Y} and B := {(x,z,y,y') \ (x,y') G grS, y,z € Y} are li-convex sets (the first being a closed linear subspace). Therefore gr(# + S) = PrXxY (THAHB) is li-convex. •

14

Preliminary

results on functional

analysis

Let Y be another topological vector space and A C X xY; we introduce the conditions (Ha;) and (Hwa;) below, where x refers to the component xeX: (Ha:) If the sequences ((xn, yn)) C A and (A„)„>! C ffi+ are such that E „ > i K = 1, E n > i ^nVn^as sum y and X)„>i A«a;n is Cauchy, then the series X^n>i ^nxn *s convergent and its sum x G X verifies {x,y)EA. (Hwi) If the sequences ({xn,yn))n>1 C A and (A n ) n >i C K+ are such that ((xn,yn)) is bounded, E n > x An = 1, £ „ > i A„2/n has sum y and ^ r a > 1 A n x n is Cauchy, then the series X^ n>1 ^« x « ^s convergent and its sum x € X verifies (x, y) 6 A. Of course, when X is a locally convex space, deleting "^2n>1 Ana;n is Cauchy" in condition (Hwa;) one obtains an equivalent statement. In the next result we mention the relationships among conditions (Ha;), (Hwi), ideal convexity, cs-closedness, cs-completeness and convexity. The proof being very easy we omit it. Proposition 1.2.6 sets.

Let A C X xY

and BcXxYxZbe

nonempty

(i) Assume that Y is complete. Then B satisfies (H.(x,y)) if and only if B satisfies (Ha;); B satisfies (Hwa;) if and only if B satisfies (Hw(x,j/)). (ii) Assume that X is complete. Then A satisfies (Ha;) if and only if A is cs-closed; A satisfies (Hwa;) if and only if A is ideally convex. (iii) Assume that Y is complete. Then A satisfies (Ha;) if and only if A is cs-complete; A satisfies (Hwa;) if and only if A is bcs-complete. (iv) If A satisfies (Ha;) then A satisfies (Hwx); if A satisfies (Hwa;) then A is convex. (v) Assume that X is a locally convex space and Prx(A) is bounded. If A satisfies (Ha;) then Pry(^4) is cs-closed; if A satisfies (Hwa;) then Pry (A) is ideally convex. We define now several interiority notions. Let 0 / A c X. We denote by rint A the interior of A with respect to aff A, i.e. rint A := intafj A A. Of

Closedness and inferiority

notions

15

course, rint A C %A. Consider also the sets c

. lA 1

.

riA:= | i6

.

-{

if aff A is a closed set, otherwise,

rint A 0

| lA

if aff A is a closed set, otherwise,

if XQ is a barreled linear subspace, otherwise,

where X0 = \in(A — a) for some (every) a G A; XQ is the linear subspace, parallel to aff A. In the sequel, in this section, A C X is a nonempty convex set. Taking into account the characterization (1.1) of *A, we obtain that x € tcA <£> cone(A — x) is a closed linear subspace of X & M

X(A — x) is a closed linear subspace of X,

and x £

A -£> cone(^4 — a;) is a barreled linear subspace of X <£> M

n(A — x) is a barreled linear subspace of X.

If X is a Frechet space and aff A is closed then %CA = lbA, but it is possible to have ibA ^ 0 and icA = 0 (if aff A is not closed). The quasi relative interior of A is the set qri A := {x € A | cone(yl — z) is a linear subspace of X}. Taking into account that in a finite dimensional separated topological vector space the closure of a convex cone C is a linear subspace if and only if C is a linear subspace (Exercise!), it follows that in this case qri A = %A = ic A = ibA Below we collect several properties of the quasi relative interior. Proposition 1.2.7 L{X,Y). Then:

Let A C X be a nonempty convex set and T £

a £ qri A •& a G A and cone(A — a) = cone(A - A) O a E i and a - A C cone(yi - a), i

A C qriA = AnqriA,

V a G q r i A , Vx £ A : [a,z[CqriA,

(1.4) (1.5)

Preliminary

16

results on functional

analysis

and T(qriA) C qriT(A). In particular qriA is a convex set. Assume that qriA ^ 0; then qriA = A and *(T(A)) C T(qriA) C qriT(A) C T(A) c T(qriA).

(1.6)

Moreover, if Y is separated and finite dimensional then T(qiiA)=i(T(A)).

(1.7)

Proof. The first equivalence in Eq. (1.4) is immediate from the definition of qriA. Of course, cone (A — a) = cone (A — A) implies that a — A C cone(A - a). Conversely, if a — A C cone(A - a) then -cone(A - a) = cone(a — A) C cone (A — a), which shows that cone (A — a) is a linear subspace. Therefore Eq. (1.4) holds. If a £ lA, from Eq. (1.1) we have that a £ A and cone(A — a) is a linear subspace, and so cone(A — a) is a linear subspace, too. Hence a € qriA. The equality qriA = A n qriA follows immediately from the relation coneC = cone C, valid for every nonempty subset C of X. Let a € qri A, x e A, A € [0,1[ and a\ :— (1 — X)a + Xx. Then A-ADA-a

A

= ( l - A)(A - a) + A(A - z) D (1 - A)(A - a),

and so, taking into account Eq. (1.4), we have cone(A - A) D cone(A - a\) D cone ((1 - A)(A - a)) = cone (A — a) = cone (A — A). Therefore cone(A — A) = cone(A — a^)- Since a\ € A, from Eq. (1.4) we obtain that a\ G qriA. The proof of Eq. (1.5) is complete. Let a £ qriA; then, by Eq. (1.4), a - A C cone(A — a), and so Ta - T{A) = T(a-A)cT = cone (T(A) -

(cone(A - a)) C T (cone(A - a)) Ta),

which shows that Ta € qriT(A). Assume now that qriA ^ 0 and fix a £ qriA. It is sufficient to show Eq. (1.6); then the equality qriA = A follows immediately from the last inclusion in Eq. (1.6) for T = Idx and from Eq. (1.5). Let y e i (T'(A)); using Eq. (1.1), there exists A > 0 such that (1 + X)y XT a € T(A). So, (1 + X)y - XTa = Tx for some x £ A. It follows that y = Txx, where xx ~ (1 + A) _1 (Aa + x). But, using Eq. (1.4), x\ 6 qriA, and so y 6 T(qriA).

Closedness and inferiority

notions

17

The second inclusion in Eq. (1.6) was already proved, while the third is obvious. So, let y G T(A); there exists x G A such that y = Tx. By Eq. (1.5), (1-X)a + Xx G qriAfor A G]0,1[, and so (l-X)Ta + Xy G T(qriA) for A G ]0,1[. Taking the limit for A —> 1, we obtain that y G T(qri A). When Y is separated and finite dimensional we have (as already observed) that i(T(A)) = qriT(A); then Eq. (1.7) follows immediately from Eq. (1.6). • The notion of quasi relative interior is related to that of united sets. Let X be a locally convex space and A,B C X he nonempty convex sets; we say that A and B are united if they cannot be properly separated, i.e. if every closed hyperplane which separates A and B contains both of them. The next result is related to the above notions. Proposition 1.2.8 Let X be a locally convex space, A,BcX empty convex sets and x G X. (i) A and B are united O- cone(A-B) is a linear subspace.

be non-

is a linear subspace •& (A — B)~

(ii) Assume that cone (A — x) is a linear subspace. Then x G c\A. Moreover, i/aff A is closed and rint A ^ 0 then ~x G rint A. Proof, (i) Assume that A and B are united but C := cone(A — B) is not a linear subspace. Then there exists XQ G (—C) \ C. By Theorem 1.1.5 there exists x* G X* such that {x0, x*) < 0 < (z, x*) for every z G C. Then 0 < (x - y,x*) for all x G A, y G B, and so (x,x*) < X < (y,x*) for all x G A, y G B, for some A G M. Therefore Hx* t\ separates A and B. It follows that (x,x*) = X = (y,x*) for all x G A, y G B, and so 0 < (z,x*) for every z G C. Thus we have the contradiction (x0,x*) = 0. Therefore cone (A — B) is a linear subspace. Let C := cone (A — B) be a linear subspace and (x,x*) < X < (y, x") for all x G A, y G B, for some x* G X* and A e i . Then 0 < (z,x*) for every z G A — B, and so 0 < (z, x*) for z € C. Then, since C is a linear subspace, 0 = (z,x*) for every z G C which implies immediately that Hx*,\ contains A and B. Therefore A and B are united. The other equivalence is an immediate consequence of the bipolar theorem (Theorem 1.1.9). (ii) By (i) we have that {x} and A are united. Assuming that x $ cl A, using again Theorem 1.1.5, we obtain that {x} and A can be properly separated. This contradiction proves that x E. cl A.

18

Preliminary

results on functional

analysis

Suppose now that aSA is closed and rint A / 0. Without loss of generality we suppose that x = 0. By what was shown above we have that x = 0 € clA C cl(affA) = aff A. Thus X0 := aS A is a linear space. Assuming now that 0 £ intx 0 A ^ 0, we obtain that {x} and A can be properly separated (in XQ, and therefore in X) using Theorem 1.1.3. This contradiction proves that x € rint A. • From the preceding proposition we obtain that when X is a locally convex space and A C X is a nonempty convex set, the quasi relative interior of A is given by the formula

qri A = A (~1 {x € X \ {x} and A are united}. The next result shows that the class of convex sets with nonempty quasi relative interior is large enough. Proposition 1.2.9 Let X be a first countable separable locally convex space and A C X be a nonempty cs-complete set. Then qri A ^ 0. Proof. Since X is first countable, by Proposition 1.1.11, the topology of X is determined by a countable family 7 = {pn \ n 6 N} of semi-norms. Without loss of generality we suppose that pn < pn+i for every n € N. Since Ty is semi-metrizable, the set A is separable, too. Let A0 = {xn | n € N} C A be such that A C cl A0. Consider j3n € ]0,2~n] such that /3npn(xn) < 2 _ n . The series ^ n > 1 Pnxn is Cauchy (since for m > n and p e N w e have that PniT^Z^Xk)

< ET=mPkPn(xk)

< ET=Z^Pk(xk)

< 2"™+!). Taking

^n '•= (Z)n>i Pn) 1Pn, ]C n >i ^ n 1 " *s a Cauchy convex series with elements of A. Because A is cs-complete, the series ^ n > 1 Xnxn is convergent and its sum x € A. Suppose that x ^ qri A. Then there exists XQ € (—C)\C, where C := cb~m(A — x). Using Theorem 1.1.5, there exists x* € X* such that (xo,x*) < 0 < {z,x*) for all z € C. In particular ( *) > 0 for — = every x £ A. But E n > i ^» ( ^ ^^o) 0- Since {xn — X,XQ) > 0 and A„ > 0 for every n, we obtain that (x - x, x*) = 0 for every x € AQ. Since AQ is dense in A and x* is continuous, we have that {x — x, x*) = 0 for every x € A, and so (z,a;*) = 0 for every z E C. Thus we get the contradiction (x0,x*) = 0. Therefore qri A ^ 0. D

Open mapping

1.3

theorems

19

Open Mapping Theorems

Throughout this section the spaces X, Y are topological vector spaces if not stated otherwise. We begin with some auxiliary results. Lemma 1.3.1 Let X,Y be first countable topological vector spaces, 31 : X =$ Y be a multifunction and XQ G X. Suppose that grIR satisfies condition (Hwa;). Then

where Nxl^o) ** the class of all neighborhoods of XQ. Proof. Let y0 G C\u£7fx(x0)int (clft(t/)). Replacing grft by gv% {x0,y0) if necessary, we may assume that (xo,yo) = (0,0). Let U G NxSince X is first countable, there exists a base (Un)n>i C N x of neighborhoods of 0 such that Un + Un C Un-i for every n > 1, where [7o := ?/• Because 0 G f " ) ^ ^ int (cl#([/)), there exists (V n ) n >i C Ny such that V„ C int (cl3i([7n)) for every n > 1. Since Y is first countable, we may suppose that (Vn)n>i is a base of neighborhoods of 0 G Y and, moreover, y n + i + Vn+i C KJ for every n > 1. Consider j / ' G int (clCR.(C/i)); there exists fi G]0,1[ such that y := (1 — lJ)~1y' £ cl3?([/i). There exists (£1,2/1) e g r $ such that x\ G C/i and 2/ - 2/i G M^2- It follows that /x -1 (y - j/i) G V2 C cl!R(t/2)- There exists (^2,2/2) G grR such that xi G L/2 and /x_1(2/ _ 2/i) — 2/2 G M^3- It follows that /U~2y — fJ-~2yi — H~lyi 6 V3 C c\"R{Uz). Continuing in this way we find ({xm,ym))m>1 C grR such that xm e Um and n~m+xy - n~m+1yi -m+2 M 2/2 - ••• - ym G fiVm+i for every m > 1. Therefore u m := y 2/1-/^2/2 At",_12/m G ^ m K i + i C V m + i for every m > 1. Moreover, m_1 M 2/m = vm-i -vm e \im~xVm - fimVm+1, and so y m £ Vro - fiVm+1 C Vm + Vm C V m _i for m > 2. It follows that (x m ) m >i -> 0, (y m ) m >i -> 0 and (u m ) m >i -> 0, whence J2m>i nm~lym has sum y. Taking Am := ( l - / i ) ^ m _ 1 , we have that (A m ) m >i C R)., E m > i A™ = *> E m > i Am2/m h a s sum (1 - ju)y = y', the sequence ((x TO ,y m )) is bounded (being convergent) and, because £mtf„ + i Ama;TO € C/„+i + Un+2 + ••• + Un+P C f7„ +1 + Un+1 C f/n for every n > 1, the series 5Z m > 1 A m x m is Cauchy. By hypothesis there exists x' £ X, sum of the series ]Cm>i Am#m> such that (x',y') £ g r X Let x := (1 - n)~lx'- We have that YZi=i Hm~lxm G t/i + U2 + • • • + Un C t/i -I- Ui C t/. Since U is closed we have that x E.U, and so a;' G (1 — n)U C

20

Preliminary

results on functional

analysis

U. Thus y' £ "R{U). Therefore int (cl3l(l7i)) C R(U). 0 € int%(U). The proof is complete.

In particular •

Note that we didn't use the fact that X or Y is separated. Observe also that it is no need that x0 £ dom"R when y0 £ f\ue^x(xo) * nt ( c ^ ( ^ 0 ) ' but, necessarily, xo £ cl(domD?) and yo £ int(ImlR) in our conditions. Note also that condition (Hwa;) may be weakened by asking that the sequence ((xm,ym)) C A be convergent instead of being bounded. In the case of normed spaces one has the following variant of the result in Lemma 1.3.1. Having the normed vector space (nvs) (X, ||-||), we denote by Ux the closed unit ball {x £ X \ \\x\\ < 1}, by Bx the open unit ball {x € X | ||a;|| < 1} and by Sx the unit sphere {x 6 X | ||a;|| = 1}. Lemma 1.3.2 Let (X, || • ||), (Y, || • ||) be two normed linear spaces and let 31: X =4 Y be a multifunction. Suppose that condition (Hwa;) holds and (x0,y0) &X xY. If yo + nUy C cl (^(XQ +

pUx)),

where rj,p > 0, then yo + nBy C%(x0 + pUx). Proof. We may take (a;o,2/o) = (0,0). One follows the same argument as in the proof of the preceding lemma, but with Un := pUx and Vn := nBy for n > 1. Consider y' 6 nBy and take p, £]0,1[ such that y :— (1 — p) y' £ c\"R(pUx)- We find the sequence {{xn,yn))n>l C grft such that (xn) C pUx and v„ := y - 2/1 - py2 ^ " _ 1 2 / n £ pnr)BY for n > 1. _1 Hence (u n ) -> 0 and /x" 2/„ = u„_i - vn, whence \\yn\\ < 77(1 + p) for n > 1. Taking An := (1 — p)pn^1 > 0 for n > 1, X) n >i ^" = 1> * n e s e r i e s S « > i ^n%n is Cauchy and the series X3n>i ^n2/n is convergent with sum y'. Since 51 satisfies the condition (Hwi), we obtain that the series J2n>i ^nXn is convergent with sum x' and (x',y') £ gr!R. Of course, x' £ pUx- Hence

nBy C npUx)-

•

Lemma 1.3.3 Let X,Y be topological vector spaces, 3? : X =4 Y be a closed convex multifunction and XQ € X. Suppose that X is complete and first countable. Then condition (1.8) holds.

Open mapping

theorems

21

Proof. Let y0 G f)u£Nx(x0)int ( cl %( U ))- Replacing gr51 by g r R - { x 0 , y0) if necessary, we may assume that (xo,yo) = (0,0). Let us first show that V i £ ] 0 , l [ , VU,U' £Kx

• tc\3l(U)c

31 {t{U + U')).

(1.9)

Fix t G ]0,1[ and take tn := t2 for n > 1. Of course, lim£„ • • • t\ = t. Consider U' G Kx- There exists a base of neighborhoods (t/„) n eN C K x such that U\ +U\ C U' and Un+i + Un+i C Un for every n € N. Let Un := (1 - i „ ) - 1 i „ • --tyUn- For every n G N there exists Vj( 6 Ky such that V^ C cl#([/'„). Consider V„ := ^ ( l - t„)V n . Let £/ G K x and j/ G cl %{U); we intend to show that ty G 31 (*(£/ + £/'))• We construct a sequence x = XQ G U, X\ G t/i, . . . , xn G J7 n ,... with the property: V n G N : t n .. .hy G clK (i„ .. .tx (x + ••• + xn-i

+Un)).

(1.10)

Since (t/ - Vi) n 3?([7) ^ 0, there exist yx G Vi and x G J7 such that y — yi G 3£(a:); hence tij/ = *i(y - yi) + (i - *0 ((i - hyhm) C ti3i(a;) + (1 - t i ) c l K ( ^ ) C c\5t{h(x

G t&ix)

+ (l -

W

+ Ui)).

Suppose that we already have x £ U, xi £ Ui,... ,xn-i desired property. Since

G £/n-i with the

(i„ . . . tiy - Vn+i) n 31 (i„ . . . h(x + • • • + x„_i + Un)) + 0, there exist yn+i G Vn+i

an

d xn G C/n such that

tn---hy-yn+i

G3i(tn...ti(H

F x „ - i + xn)).

Therefore tn+l •••hy

= tn+l(tn-

--hy

— 2/n+l) + ( 1 - < n + l ) ( ( 1

£ tn+i$.(tn...ti(x-{

ha;„_! +xn))

_

tn+1)~~

+ (1

tn+1yn+i)

-tn+1)V„+1

C i„+i^(*n .. . t i ( i + • • • + a; n -i + a;„)) + (1 - t„+i) d 3 l ( t / ; + 1 ) C cl3t(t n +i ...h(x-\

hx„_i +xn +

Un+i)).

So the desired sequence is obtained. Since xn+\ H h x „ + p G Un+i + • • • + Un+P C Un for all n,p € N, the series 2 n > i x™ ' s convergent. Denote by x1 its sum. Since i i + - + i » E i i + f/i and Ui is closed we have that x1 Gxi+Ux C U'.

22

Preliminary

results on functional

analysis

Let now U" E Nx and V" E Ny be arbitrary. There exists n E N such that tn ...h(x-\ \-xn-i+Un) C t(x+x')+U" andtn .. .hy G ty+V". By Eq. (1.10) we obtain that tn ... txy E cl!R(t(x + x') + U"). It follows that (ty + V")nJi(t{x + x') + U") ^ Hi, and so there exist x" E U" and y" G V" such that ty + y" E $.(t(x + x')+x"), i.e. (t(x + x'),ty) + (x",y") € g r # . It follows that (t{x + x'), ty) E cl (gr 3V) = grft. Therefore ty E ft (t(U + U')). To complete the proof, let U E Nx(0). Then there exists U' E Nx such that [/' + [/' C 1U. From Eq. (1.9) we obtain that | c l # ( E / ' ) C 3? ( | ( [ / ' + [/')) C #(*/). Since 0 E C\ue^x{xo) int (cl3i(U)), we have that 0G int£([/). ° D Note that Lemma 1.3.3 is a particular case of Lemma 1.3.1 when Y is first countable; otherwise these results are independent. Corollary 1.3.4 Let Y be a first countable topological vector space and C CY be an ideally convex set. Then intC = int(clC). Proof. Consider X an arbitrary Frechet space (for example X = R) and ft : X =t Y denned by IR(0) = C, R(x) = 0 for x ^ 0. Of course condition (Hwir) is satisfied. Taking y0 E int(clC) and xo = 0 we have that 2/o G f)ue^x(xo)int ( c l 3 i ( C 7 ))' a n d s o 2/o € int C. O Theorem 1.3.5 (Simons) Let X and Y be first countable. Assume that X is a locally convex space, "R : X =$ Y satisfies condition (Hwa;), yo E lb (ImJi) and x0 E 3l_1(j/o)- Then yQ E int af f(i m s) R(U) for every U E N x ( z 0 ) . In particular i6(ImIR) = rint(Imft) if i6 (ImK) ^ 0. Proof. Once again we may consider that (xo,yo) = (0,0); so Y0 := aff(ImCR) — lin(Im3i). Replacing, if necessary, Y by YQ, we may suppose that Y is barreled and 0 G (ImIR)\ Let U E Ncx; since grft is a convex set and R(U) = Pry (gr ft n U x Y), %(U) is convex, too. "R{U) is also absorbing. Indeed, let y E Y. Because Imft is absorbing, there exists A > 0 such that Xy E Imft. Therefore there exists x E X such that (x,Xy) E grft. Since U is absorbing, there exists fi G]0,1[ such that fix E U. As grft is convex we have that (fj,x,fj,Xy) = fx(x,Xy) + (1 — /x)(0,0) G grft, whence /xAj/ G R(U). Therefore $.(U) is also absorbing. It follows that cl (£([/)) is an absorbing, closed and convex subset of the barreled space Y. Therefore 0 G int (clft(C/)). Using Lemma 1.3.1 we obtain that 0 G intft(Z7) for every neighborhood U of 0 G X. The last part is an immediate consequence of what was obtained above. •

Open mapping

theorems

23

Corollary 1.3.6 Let X be a Frechet space, Y be first countable and 51 : X =$ Y be li-convex. Assume that yo € t6(Im!R) and XQ 6 5i~1(yo)Then y0 £ intaff(im3j) 5l(U) for all U E Nx(zo)- In particular *b(Im5l) = rint(Im^) provided i6 (ImK) ^ 0. Proof. There exist a Frechet space Z and an ideally convex multifunction S : Z x X =4 Y such that grIR = P r x x y ( g r S ) - Then § verifies condition (Hw(z,i)) by Proposition 1.2.6 (ii). Of course, there exists ZQ 6 Z such that yo £ S(zo,zo)- Since ImS = Imft, by the preceding theorem, j / 0 € intaff(imX) #(E0 = intaff(imK) HZ x ^ ) for every U G Kx(^o)• Theorem 1.3.7 (Ursescu) Let X be a complete semi-metrizable locally convex space and 51 : X =£ Y be a closed convex multifunction. Assume that yo £ t6(ImCR) and xo € 5l~1(yo)- Then yo € int a ff( Im ^) 5l(U) for every U £ Xj(xo). In particular i 6 (Im^) = rint(ImK) if ib{Im 51) ^ 0. Proof. If Y is first countable it is obvious that the conclusion follows from Simons' theorem. Otherwise the proof is exactly the same as that of Simons' theorem but using Lemma 1.3.3 instead of Lemma 1.3.1. • An immediate consequence of Theorem 1.3.5 is the following corollary. Corollary 1.3.8 Let Y be a first countable barreled space and C CY be a lower ideally convex set. Then Cl = intC. Proof. Let yo G Cl. There exist a Frechet space X and an ideally convex multifunction 51: X =3 Y such that C = Im 51. The conclusion follows from Theorem 1.3.5 taking again U — X. O In fact the conclusion of the above corollary holds if C is the projection on Y of a subset A of X x Y with (a') X is a first countable locally convex space and A satisfies condition (Hwx) or (b') X is a semi-metrizable complete locally convex space and A is closed and convex. Putting together Corollaries 1.3.4 and 1.3.8 we get the next result. Corollary 1.3.9 Let Y be a first countable barreled space and C C Y be an ideally convex set. Then Cl = intC = int(clC) = ( c l C ) \ • In normed spaces the following inversion mapping theorem holds. T h e o r e m 1.3.10 (Robinson) Let (X, ||-||) and (Y, ||-||) be normed vector spaces, 51 : X =S Y be a convex multifunction and (xo,yo) € g r ^ - Assume

Preliminary

24

results on functional

analysis

that y0 + rjUy C 3\{XQ + pUx) for some n,p > 0. Then d f c . K - 1 ^ ) ) < P+\\x-x0\\ v-\\y-yo\\

,d(

^ ( a .))

VarGX, Vy£y0

+ r,BY.

Proof. Replacing g r # by g r # — (xo,yo), we may assume that (xo, yo) = (0,0). Let x E X and y G nBy. The conclusion is obvious if x £ domIR or y € &(z), so suppose that neither is true. Choose 6 > 0 and find ye € 3£(:c) such that 0 < \\ye - y\\ < d[y, R(x)) + 6; define a := n - \\y\\ > 0 and take e G ]0, a[. Consider ye :=y + {a-e)\\y-ye\\~1

(y - ye);

thus ||y £ || < \\y\\ + (a — s) =n — e, and so ye G nBy- Therefore there exists xe G pUx with y£ G R(x€). Define A := \\y -yg\\(ae + \\y - y ^ l ) - 1 G ]0,1[. Then 2/ = (1 - X)ye + Xys G (1 - A)tt(a:) + A#(x £ ) C E((l - A)z + Aa;£). Thus (1 - A)z + Xx£ G 3£_1(j/), whence d ( x , ^ - 1 ^ ) ) < A | | x - x e | | . As ll^ — ^ell < \\x\\ + ll^ell < P + INI a n d A < (a — e ) _ 1 ||y — yg\\, we obtain that d (x^iy))

<(P+

\\x\\) (a - e ) ' 1 (d(y, X(x)) + 9).

Letting 6, e —> 0, we obtain the conclusion.

•

Combining the preceding result and Lemma 1.3.2 we obtain the following important result in normed spaces. The implication (i) =$> (ii) is met in the literature as the Robinson-Ursescu theorem. Theorem 1.3.11 Let (X, ||-||), (Y, ||-||) be two normed vector spaces and 31 : X z t 7 be a multifunction. Suppose that Y is a barreled space, g r # verifies condition (Hwx) and (xo,yo) G gr3i. Then the following conditions are equivalent: (i)

yoeQmX)';

(ii) 2/0 e int:R(xo + Ux)\ (iii) 3n > 0, VA G [0,1] : 2/0 + XnUy C IR(a;o + At/ X ); (iv) 37,77 > 0, Vz G z 0 + r?t/ X; Vy G y0 + nUy : d ( ^ S T ^ y ) ) < 7-d(y,K(a;)); (v)

3 r ? > 0 , V y G y o + r ? [/x, V x G X

d(y,3i(i)).

: d(x,-R-\y))

< ^Efff •

Open mapping

theorems

25

Proof. The implications (hi) =>• (ii) =>• (i) are obvious. The implication (i) =$> (ii) follows immediately from Simons' theorem, while the implication (ii) =$> (v) is given by the preceding theorem. (iv) => (iii) Let 7' > 7; we may assume that 777' < 1, because, in the contrary case, we replace 77 by 77' := I / 7 ' < 77. Let y £ yo + A?7?7y, y 7^ j/o, w i t h A e ] 0 , l ] . Thend(a;o,^- 1 (y)) < 7 • d{y,3L(x0)) < 7' ||y - yQ\\. Hence there exists x € 5l~1(y) such that ||x —io|| < 7'|l2/~2/o|| < 7'Ar/ < 77. Therefore y0 + A^C/y C R(x0 + \UX)(v) => (iv) Taking a; € XQ + §t/x, 2/ G j/o + f^y' w e obtain that d (x, ft"1 (y)) Y be a linear operator. Then T is continuous if and only if gr T is closed in X x Y. Proof. It is obvious that gr T is closed if T is continuous (even without being linear). Suppose that g r T is closed and consider the multifunction 51 := T _ 1 : Y =$ X. It is obvious that grlR is closed and convex (even linear subspace). Moreover ImO? = X. So we can apply Theorem 1.3.5 for (Tx0,x0) G Y x X. Therefore

VVeNHTio) :

R(V)=T-1(V)£Nx(x0),

which means that T is continuous at XQ.

•

Corollary 1.3.13 (Banach-Steinhaus) Let X, Y be Frechet spaces and T : X —> Y be a bijective linear operator. Then T is continuous if and only if T _ 1 is continuous; in particular, if T is continuous then T is an isomorphism of Frechet spaces. Proof.

Apply the closed graph theorem for T and T - 1 , respectively. D

Theorem 1.3.14 (open mapping theorem) Let X, Y be Frechet spaces and T G L(X,Y) be onto. Then T is open. Proof. Of course, T is a closed convex relation and Im T = Y. Let D C X be open and take 7/0 G T{D)\ there exists XQ G D such that 7/0 = Tx0. Applying the Ursescu theorem for this point, since D is a neighborhood of XQ, we have that T(D) is a neighborhood of T/0. Therefore T(D) is open.D

Preliminary

26

results on functional

analysis

An interesting and useful result is the following. Corollary 1.3.15 Let X, Y be Frechet spaces, A C X and T : X ->• Y be a continuous linear operator. Suppose that I m T is closed. Then T(A) is closed if and only if A + ker T is closed. Proof. Replacing, if necessary, F by I m T and T by T" : X -> I m T , T'{x) := T(x) for x G X, we may suppose that T is onto. Consider T : Xj kerT —> Y, T(x) := T(x), where x is the class of x. It is easy to verify that T is well defined, linear and bijective. Let q : X —> M. be a continuous semi-norm; since T is continuous, p := qoT is a continuous semi-norm, too. For all x € X and u € ker T we have that ( X/keiT, ir(x) := x, we obtain that T(A) is closed & n(A) - A is closed <$• IT'1 (A) = A + ker T is closed, and so the conclusion holds.

•

As an application of the Simons and Ursescu theorems we give the following two interesting results, useful in studying optimal control problems. We recall that a process is a multifunction C : X => Y whose graph is a cone; when the graph of the process C is convex or closed one says that e is a convex process or closed process, respectively. The adjoint of the process 6 is the w*-closed convex process 6* : Y* =3 X* whose graph is the set {(y*,x*) £ Y* x X* | (-x*,y*) € (gre)+). Theorem 1.3.16 Let X, Y, Z be Banach spaces, 6 : X =$Y be an ideally convex process and T G £(•£, Y). Consider the following statements:

(i) ImT d m 6 ; (ii) 3 P l > 0 , V(»*,*•) GgrC* : \\T*y*\\ <

(hi) 3P2>o

: T{UZ) c

Pi\\x*\\;

P2e(uxy,

(iv) 3p3>0 : T(Uz)Cp3c\(e(Ux)). Then (i) «• (hi), (ii) •£• (iv) with p\ = p3 and (hi) => (iv) with p3 = p2. Moreover, ifT is onto then (iv) => (hi) with (any) p2 > p3. Proof. The implications (hi) =$• (i) and (iii) => (iv) (with p3 = p2) are obvious.

Open mapping theorems

27

(i) => (hi) Let 3? := T~1oQ; one obtains immediately that 3J is an ideally convex process with ImIR = Z. Applying the Simons theorem (Theorem 1.3.5) for (a;o,2/o) = (0)0); there exists p > 0 such that pUz C 3£(t/x), which means that pT(Uz) C G(Ux)- Taking P2 := p"1, (hi) holds. (iv) => (ii) Let (y*,x*) G grC*, i.e. (-z*,y*) 6 (grC)+. Then | | T V I I = sup ( z , - T V > = sup ( T z , - y * ) < sup fo.-y*) ||z||
sup (y, -y*) < p3 sup{(a;, -x*) \ (x,y) G grC, ||a;|| < 1} yee(ux)


because (y, —y*) < (x, —x*) for (x,y) G grC. So (ii) holds with p\ = p3. (ii) =*• (iv) Suppose that (iv) does not hold and take z G Uz such that Tz $ /93cl(C([/x))- Since C(f/x) is convex, using Theorem 1.1.5, there exist y* G Y* and A G E such that %T*T)

= (Tz,y*) < A < (p3y,y*)

Vy G G(UX).

It follows that A < 0, and so we can take A = -p3. Hence —p3 > (z, T*y*) > -||r*y*ll and - 1 < (y,^) = (x,0) + (y,y*) for (x,y) G grC with x G Ux, whence ||T*y*|| > p3 and (0, y*) G (gr 6 n Ux x 7 ) ° = «;* - cl ((gr 6)+ + tf*. x {0}) = (gre)+ + ?7x' x { 0 } because Ux* x {0} is W*-compact (we have used the Alaoglu-Bourbaki theorem). Therefore there exists x* G Ux* such that (y*,x*) G gr(2* (<S> ( - r ,j/*) G (grC)*). Since \\T*y*\\ > p3 > p3 \\x*\\, (ii) does not hold for pi= p3. (iv) =^ (iii) when T is onto. Since T is onto and Z, Y are Banach spaces, T is open (see Theorem 1.3.14). It follows that int(T(Uz)) = T{BZ). But G(Ux) is cs-closed. Indeed, by Proposition 1.2.2, A := grCnUx x Y is cs-closed, and so, X being complete, A satisfies condition (Hz) (see Proposition 1.2.6(h)); by Proposition 1.2.6(v) we have that Pry(A) = G(Ux) is cs-closed. Using Corollary 1.3.4 we obtain that intC([/x) = iat(c\G(Ux))So, from (iv) we obtain that T(BZ) C p3Q(Ux), and so T{UZ) C p p3. D A similar result holds in dual spaces.

Preliminary results on functional analysis

28

T h e o r e m 1.3.17 Let X,Y,Z be normed spaces, T e £>{X,Y) and 6 : X =$ Z be a convex process. Consider the following statements: (i)

ImT'Clme*;

(ii) 3 p > 0 , V(x,z) (iii) 3 p > 0 :

S g r C : ||Ta;|| < p||;z||; T*(UY*)CPe*(Uz>).

Then (i) O- (iii) and (ii) O- (iii) with the same p. Proof. The equivalence of (i) and (iii) follows from the equivalence of (i) and (iii) of the preceding theorem; just apply it for the Banach spaces X*, Y*, Z*, the continuous linear operator T* and the closed convex process

e*. (iii) =>• (ii) The proof follows the same lines as the proof of (iv) =£• (ii) in the preceding theorem, so we omit it. (ii) => (iii) Even if the proof is similar to (ii) =>• (iv) in the preceding theorem, we give the details. So, suppose that (iii) does not hold and take y* e Uy* such that T*y* £ pG*(Uz>). We may assume that y* £ Sx(otherwise replace y* by ||2/*|| _1 y*). Using the fact that Uz* is w*-compact and gre* is u>*-closed it follows easily that G*(Uz*) is uAclosed; being also convex, we can apply Theorem 1.1.5 in (X*,w*). Therefore there exist x £ X and A G E such that (Tx,y*) = {x,T*y*) > A > (x,px*)

Vx* € e*{Uz>).

Hence A > 0, and we can take X = p. Hence ||Tx|| > p and 1 > (x, x*) for every x* € e*(Uz>), i.e. - 1 < (x,x*) + (0,z*) for (x*,z*) 6 (grC)+ with z* £ Uz* • It follows that (af,0) € ( ( g r Q + f l l x Uz.)° = cl ((gre)++ + {0} x UZ)

=

cl(gve+{0}xUz).

Hence there exist ((xn,zn)) C grC and (z'n) C Uz such that (xn) -> x and (wn) := (zn + z'n) ->• 0. It follows that (||Ta;„||) ->• \\Tx\\ and \\zn\\ = \\wn — z'n\\ < 1 + ||iwn|| for every n, whence plimsup \\zn\\ < p < \\Tx\\ < liminf ||Ta;„||. Therefore there exists n 0 such that ||Ta; n || > p||z„|| for n >n0, which proves that (ii) does not hold for the same p. O

Variational

1.4

principles

29

Variational Principles

A very important result in nonlinear analysis is the "Ekeland variational principle." Theorem 1.4.1 (Ekeland) Let {X,d) be a complete metric space and f : X —» M. be a proper lower semicontinuous and lower bounded function. Then for every xo G dom / and e > 0 there exists xe G X such that f(xE) < f(x0)

-ed(xQ,xE)

and f{xe)
Vx6l\{x£}.

Proof. Let XQ G dom / and e > 0 be fixed. For every x G X consider the set F(x) := {y G X \ f{y) + ed(x,y) < f(x)}. Note that the conclusion is equivalent to the existence of an xe G F(xo) such that F(xs) = {xE}. Since the function X 3 y *-> f(y) + ed(x, y) G E is lower semicontinuous (lsc for short), F(x) is closed. Note that x G F(x) C d o m / for every x G d o m / and F(x) = X for x G X \ d o m / . Also note that F(y) C F(x) for every y G F(x). The inclusion is obvious for x ^ d o m / . So, let x G d o m / , y G F(x) and z G F(y). Then f{z)+sd(y,z)
f{y)+sd(x,y)

< f{x),

d(x,z) < d(x,y) + d{y,z).

Multiplying the last relation by e, then adding all three relations we obtain that f(z) + ed(x,z) < f(x), i.e. z G F(x). Since / is bounded from below, s0 := inf{/(x) | x G F(xo)} G E; take X\ G F(xo) such that / ( ^ i ) < SQ + 2~1. Then consider si := inf{/(z) | x G F(xi)} G E and take x2 G F(a;i) such that ffa) < s\ + 2~2. Continuing in this way we obtain the sequences (sn)n>o C M. and (xn)n>o C X such that sn = inf{/(a;) | x G F(xn)},

xn+1 G F(xn),

for all n > 0. Because i^(a; n+ i) C F(xn), as xn+i G F ( i „ ) , ed(a; ri+ i,a; n ) < f(xn)

- f(xn+1)

f{xn+1)

< sn + 2 - " - 1

s„+i > s„ for n > 0. Moreover,

< f{xn) - sn < f(xn)

- s„_i < 2~ n

for n > 1, whence d(x„ +p ,a;„) < e~121~n for n , p > 1. This shows that (xn) is a Cauchy sequence, and so (xn) converges to some xs G X. Since

30

Preliminary

results on functional

analysis

xm G F{xn) for m > n, we have that xE € clF(a; n ) = F(xn) for every n > 0. In particular xE £ F{XQ). Let a; 6 F(ar £ ); then x € F(xn) for every n > 0, and so, as above, ed(x,xn) < f(x„) — f(x) < f(x„) - sn < 2~n, which shows that (xn) —>• x. As the limit is unique, we get x = xe, which shows that F{xe) = {xe}. The proof is complete. • In applications one often uses the following variant of the Ekeland variational principle. In the sequel by infx / , or simply inf/, we mean mi{f(x) | x € X}, where / : X -> I . Corollary 1.4.2 Let (X,d) be a complete metric space and f : X -» R be a bounded below lower semicontinuous proper function. Let also e > 0 and xo S d o m / be such that f(xo) < mix f + £• Then for every A > 0 there exists x\ S X such that f(x\)

< f(x0),

d(xx,x0)

< A

and f(xx)

< f{x) + eX-idfaxx)

Va; £ X \

{xx}.

Proof. Applying the preceding theorem for xo and eA _ 1 , we get x\ € X satisfying the second relation of the conclusion and f(x\)+e\~1d(xo,x\) < f(xo). Hence f{x\) < f{xo). Moreover, because f(x0) < inf^ / + £ < f(x\) + e, we get also that d{x\, XQ) < A. D A good compromise is obtained by taking A = -Jz in the preceding result. An interesting application of the Ekeland theorem is the following result. Corollary 1.4.3 Let (X, ||-||) be a Banach space and f : X -> R be a lower semicontinuous function. Assume that f is Gateaux differ•entiable and bounded from below. If (xn)n^ is a minimizing sequence for f, i.e. (f(xn)) -> infjt / , then there exists a minimizing sequence (xn) for f such that (\\xn -xn\\) -> 0 and (||V/(x„)||) -» 0. Proof. Consider en := f(xn) — infx f £ K+- If f{xn) = infx f we take xn := xn; otherwise, applying the preceding corollary for xn, en and A = y/e„~ we get xn e X such that f(x„) < f(xn), \\x - xn\\ < ^fe^ and f(xn)
MxeX.

Note that these relations hold also in the case e„ = 0 (with our choice). Taking x := xn + tu with t > 0 and u £ X, we get -y/s^ \\u\\ < t~l(f(xn +

Variational

principles

31

tu) — f{xn)). Taking the limit for t —¥ 0 we get — y/e^\\u\\ < (u,S7f(xn)) for all u £ X. It follows that ||V/(z„)|| < y/e^. The conclusion follows. • The next result is met in the literature as the smooth variational principle. This name is due to the fact that, at least in Hilbert spaces, the perturbation function is smooth (for p > 1). The result will be stated in a Banach space (X, ||-||) even if it holds in any complete metric space (with the same proof). Before stating it let us observe that, when (un)n>o C X is bounded, (nn)n>o C M+ is such that Yln>o ^n — 1 a n ^ P £ [1J °°[> the function

Qp : X -+ E, Qp(x) := ^2n>0^n ^ ~ Un^ '

(L11)

is well defined and Lipschitz on bounded sets; in particular 0 P is continuous. Theorem 1.4.4 (Borwein-Preiss) Let (X, ||-||) be a Banach space, f : X —> M. be a proper lower semicontinuous and bounded from below function and p 6 [1, oo[. If xo £ X and e > 0 are such that f(xo) < mix f + £ then for every A > 0 there exists a sequence (wn)n>o C B(xo,X) converging to some u € X such that f(u) < mix f + e,

\\xo - u\\ < A

and f(u) + e\-pQp{u)

< f(x) + e\-pQp{x)

VxeX,

(1.12)

where Qp is defined by Eq. (1.11). Proof.

Fix A > 0 and consider 7, S, n, /i, 6 > 0 such that

6 < M ( l - (7 ? 7~ 1 ) 1 / T> (1.13) p and 6 := (1 - fi)e\~ . Let uo := #o and / 0 := / . Taking fi(x) := fo(x) + 5 \\x - u0\\p for a; £ I , we have that fi(u0) = /o(«o) > mix h- Hence there exists u\ 6 X such that f\(u\) < 6f0(u0) + (1 — 6) mix fi- Taking f2(x) := fi(x) + 5/i\\x — ui\\p for 1 £ I , we have that /2(ui) = fi(ui) > mix fiHence there exists u2 £ X such that f2(u2) < 6f\(u\) + (1 — 0)infx / 2 . Continuing in this way we obtain a sequence (u„) n >o C X and a sequence of functions (fn)n>o such that f(x0) - mixf

< V < 7 < e,

fn+i(x)=fn(x)+6fin\\x-un\\p

H
Vx£l,Vn>0,

(1.14)

32

Preliminary

results on functional

analysis

and fn+l{Un+l)
Vn>0.

(1.15)

It is clear that / „ < fn+i; taking sn := infj*:/ n , (s„)„>o C E is a nondecreasing sequence. Let an := / „ ( u n ) ; because fn+i{un) = fn(un) > infx/n+i, from Eq. (1.15) we obtain that an+\ < an for every n > 0. Using again Eq. (1.15), we get sn < s„+i < an+i < 0an + (1 - 9)sn+i

< an,

and so a-n+i - sn+i < Qan + (1 - 6)sn+i - sn+i = 6(an - sn+i) < 0(an - s„), whence an-sn<9n(a0-s0)

Vn > 0.

(1.16)

It follows that lims„ = lima„ € E. Taking x := u n +i in Eq. (1.15), we get fln > a«+i = fn{un+i) + Sn" \\un+1 - un\f >sn + 6p,n ||u„+i - un\\p , which, together with Eqs. (1.13) and (1.16), yields 6n(a0 - s0) < 6nr)

Sfi" | | « n + 1 - u „ f
(77/<5) 1 / J , (6'/^)"/P,

\\un+m - u „ | | < (v/8)1/p(0/l*)n,p{l

Vn > 0.

whence

~ Wltf1')'1

Vn,m > 0.

Since 0 < #//z < 1, the sequence (u n )„>o is Cauchy, and so it converges to some u 6 X. Moreover, the inequality above and (1.13) imply that I K - unll < (V/S)1/P(vhr1/P

= (l/S)1/p

- fi)1^ = A (1.17) for all n,m > 0. In particular (un) C B(xo,X)- Letting m —> co in the above inequality we get \\u — un\\ < A for n > 0, and so \\u — xo\\ < A. Consider n„ := fin(l - fx) > 0; hence J2n>of1n = 1- Let 0 P be defined by Eq. (1.11). From Eq. (1.14) we obtain that fn+i{x)=f{x)+e\-pY.nk=0^\\x-uk\\p

< (e/Sy/p(l

Vn>0, V x e l ,

33

Variational principles and so f(x) + eX pQp(x) — lim/„(a;) > lims„. Using Eq. (1.17) we get f(un) + e\~pQp(un)

= fn(un)

+ e\~p Y]

^k \\un - uk\\p fikX" = an + e}

ak

*— / k=n+l

K=n+1

for every n > 1. Taking into account the continuity of 0 P and the lower semicontinuity of / , the preceding inequality yields f(u) + eX~pQp(u) < lima„ = lims„. Therefore (1.12) holds. From Eq. (1.17) we obtain that Qp{x0) < Y " ^Un I K - Un\\P < J~] ^—'n>l

HkjS"1 = fJ.jS-1 < fJ,XP,

^—'n>l

and so, by Eq. (1.12), f{u) < f(x0) + eX~pep{x0)

< f(x0) +sfi<

infx / + 7 + £/* < infx f + £-

The proof is complete.

D

Note that |©i(:r) -Qi(y)\ < \\x — y\\, and so Eq. (1.12) becomes f(u) < /(a;)+£A-1||a;—w|| for every 2; € Xwhenp= 1. So, under the slight stronger condition f(x0) < mix / + £ one recovers the conclusion of Corollary 1.4.2, but the condition f(x\) < f(xo). Although not directly related to what follows, we give the next two dual interesting results which have not been published by their author, C. Ursescu. Theorem 1.4.5 Let (X,d) be a complete metric space and (F„) n eN be a sequence of closed subsets of X. Then

cl

(U„ e N i n t F ")= d i n t (U„ e N ^)-

d-18)

Theorem 1.4.6 Let X be a complete metric space and (Dn)ne^ sequence of open subsets of X. Then

int

(n„ 6 N c i ^) = i n t c i (n„ e N ^)-

be a

(LI9)

Note that Theorem 1.4.5 can be obtained immediately from Theorem 1.4.6 by taking Dn = X\Fn (and Theorem 1.4.6 is obtained from Theorem 1.4.5 by taking Fn = X \ Dn). We give only the proof of Theorem 1.4.6. Proof. Since the inclusion "D" in Eq. (1.19) is obvious, we show the converse one. So let x G int (|"| n e N clD n ) and r > 0; take xi := x.

34

Preliminary

results on functional

analysis

Then there exists ri G]0,r[ such that D{xi,r\) C f]n£Nc\Dn; therefore D(xi,r\) C clD„ for every n £ N. It follows that B(x\,ri) n D\ is an open nonempty set. There exist X2 € X and r2 6]0,ri/2] such that D(x2,r2) C B(xi,n) D £>i. It follows that D f o , ^ ) C D(a;i,ri) C cl£>2) and so B{x2,r2) fl £>2 is an open nonempty set. Continuing in this way we find the sequences (a?n)neN C X and (rn)n£N C F such that (r„) n£ m ->• 0 and D(xn+\,rn+i) C B(x„,r„) n £>„ for every n. In particular D(a; n +i,r n +i) C D(xn,r„) for every n. Since X is a complete metric space, using Cantor's theorem, (~}neND(xn,rn) = {x'} for some x' € X. It follows that x' £ £>„ for every n. Since x' 6 D(xi,r{) C B(x,r), we have that B(x,r)nf\n€JiDn. As r > 0 is arbitrary, x £ cl (finer*-^M- Therefore int ( f | n e N c l D „ ) C cl ( f | n € N A i ) , whence the inclusion "C" in Eq. (1.19) holds, too. • From Theorem 1.4.5 we obtain immediately the famous Baire's theorems: Theorem 1.4.7 (Baire) Let (X,d) be a complete metric space. Then any countable intersection of dense open subsets of X is dense. Proof. Let A = C\n€N Dn with Dn open and dense for every n £ N. From Eq. (1.19) it follows that X = int(clA), and soc\A = X. • Remind that an intersection of a countable family of open subsets of the topological space (X, r ) is called a Gs set. Theorem 1.4.8 (Baire) If (-Fn)ngN is a sequence of closed subsets of the complete metric space (X,d) such that X = \Jne^Fn, then at least one of Fn 's has nonempty interior. Proof.

1.5

The conclusion is immediate from Eq. (1.18).

•

Exercises

Exercise 1.1 (Caratheodory) Let X be a linear space of dimension n 6 N and let A C X be nonempty. Prove that co A

= { Y^ill^

| M i e W T C R+, (xi) ieT ^+T C A, Y^ill^

= !} •

Exercise 1.2 Let X be a finite dimensional normed space and A C X be a nonempty convex set. Prove that 'A =£ 0, "(cl A) — 'A, cl(*A) = cl A and for every a € 'A and z g cl A we have }a, z[ C lA.

Exercises

35

E x e r c i s e 1.3 Let X be a separated locally convex space, H C X be a closed hyperplane and A C X be a nonempty convex set. If int# (A O H) ^ 0 and A <£ H prove that int ^4 ^ 0. Moreover, if M C X is a closed affine set with finite codimension and intM{A f~l M) ^ 0, prove that rint A ^ 0. E x e r c i s e 1.4 Let X be a separated locally convex space and A, C C X be nonempty convex sets with int C ^ 0. Prove that int(A + C) = A + int C. Moreover, if A n int C ^ 0, then cl(A DC) = cl yl l~l cl C. In particular, if C is a convex cone with nonempty interior, then cl C + int C — int C. E x e r c i s e 1.5 Let X be a separated locally convex space, Xo C X be a linear subspace, a i , . . . , ap G X, ipi,..., tpk G X" and a i , . . . , a* G R, where p, A; G N. Prove that: (a) if Xo is closed and C = { E L i - ^ I Vi G L7p : A, > 0}, then X 0 + (7 is a closed convex cone. In particular C is a closed convex cone; (b) Xo + {x £ X | Vi 6 1, k : (x, ipi) < at} is a convex closed set. E x e r c i s e 1.6 Prove that

Let (X, (• | •)) be a Hilbert space, xo £ X \ {0} and a G [0,7r/2]. P ( a ) := {x £ X | Z(a;, x0) < a}

is a closed convex cone, where Z-(x,y) := Arccos

" . . Moreover, P(a)°

=

P(7r/2-a). E x e r c i s e 1.7 Let n £ N \ {1}, p > 0 and a i , . . . , a „ £]0,1[ be such that a i + . . . + an = 1. Consider ATp := { ( x i , . . . , x „ , x n + i ) G R n + 1 | xi,...,x„

> 0, |x n +i| < px"1 •• -a;""} •

Prove that (Kp)+ = .Ry, where p' := (pa" 1 • • • a " " ) - 1 . In particular, Kp is a closed convex cone. Exercise 1.8

Prove that P := {(x,t/,z) £ R 3 | x,z > 0 , 2 x z > t/ 2 }

is a closed convex cone and P+ = P. (P is called the "ice cream" cone.) Exercise 1.9 Let X be a normed space, ip G X* \ {0} and 0 < a < \\(p\\. Prove that the set C := {x G X | a\\x\\} is a pointed (C 0 -C = {0}) closed convex cone with nonempty interior and C+ = R+ • ((p + aU*) = R+ • D((p, a). E x e r c i s e 1.10 Let X, Y be topological vector spaces and 31 : X =t Y be a convex multifunction. Assume that there exists x G X such that int 3^(5:) ^ 0. Prove that for every (xo,j/o) G grft with yo G (Imft) 1 there exists u G X such that j/o G intK(x) for every x &}xo,u].

Preliminary results on functional analysis

36

E x e r c i s e 1.11 Let X, Y be two separated locally convex spaces and 6 : X =4 Y be a multifunction whose graph is a closed linear subspace of X x Y. Prove that domC is dense if and only if C*(j/*) is a singleton for every y* G dome*. In particular, dom (2 is dense and C(x) is a singleton for every x G dom G if and only if domC* is w*-dense and C*(y*) is a singleton for every y* € dom6*. Exercise 1.12 Let (X, d) be a metric space. Prove that (X, d) is complete if for any Lipschitz function / : X —• R+ there exists x € X such that f(x) < f(x) + d(x,x) for every x £ X. This shows that the completeness assumption in Ekeland's variational principle is essential. Exercise 1.13 Let (X, d) be a complete metric space and / : X —)• R be a proper lsc and lower bounded function. Suppose that for every x G X such that f(x) > inf / , there exists x G X\{x} such that f(x) + d(x,x) < f(x). Prove that argmin / ^ 0 and d(x, argmin / ) < f(x) — inf / for every x E X. Exercise 1.14 Let (X, d) be a complete metric space, / : X —• R be a proper lsc lower bounded function and 3£ : X =t X. Assume that for every x G X there exists y G 3l(x) such that d(x,y) + f(y) < f(x). Prove that 31 has fixed points, i.e. there exists xo S X such that xo € 3£(xo)E x e r c i s e 1.15

Let (X, ||||) be a normed space and / : X -> R. Prove that

(i) limn^n^oo f(x) = oo if and only if [/ < A] is bounded for every A G R. (ii) Assume that X = Rfc, / is lsc and limnsn^oo f(x) = oo (i.e. f is coercive). Prove that there exists x G R* such that f(x) < f(x) for every x G Rfc.

1.6

Bibliographical Notes

Section 1.1: For the notions and results on topology and topological vector spaces not recalled in this section one can consult many classical books (see [Kelley (1955); Willard (1971); Bourbaki (1964); Holmes (1975)]). The proofs of the results mentioned in this section can be found in [Holmes (1975)] or [Bourbaki (1964)]. Section 1.2: Ideally convex sets were introduced by Lifsic (1970), cs-closed sets by Jameson (1972), cs-complete sets by Simons (1990), lower cs-closed sets by Amara and Ciligot-Travain (1999), while condition (Hx) was used by Zalinescu (1992b). The properties stated in Proposition 1.2.1 are mentioned in [Kusraev and Kutateladze (1995)]. All the results concerning lcs-closed sets, as well as Proposition 1.2.2 (for cs-closed sets), are from [Amara and Ciligot-Travain (1999)]. The set *CA, introduced by Zalinescu (1987), is also introduced by Jeyakumar and Wolkowicz (1992) under the name of strong quasi relative interior of A. The notation lbA was introduced in [ZalinescuN(1992b)] but the condition 0 G lbA was intensively used by Simons (1990) and Zalinescu (1992b). The quasi relative interior was introduced by Borwein and Lewis (1992); they stated the properties mentioned in Proposition 1.2.7 in locally convex spaces. Proposition 1.2.9 is

Bibliographical notes

37

stated in [Borwein and Lewis (1992)] for A a cs-closed set and X a Frechet space. Joly and Laurent (1971) introduced the notion of united sets, but Proposition 1.2.8 is mainly established by Moussaoui and Voile (1997). Section 1.3: Lemma 1.3.1 is proved by Amara and Ciligot-Travain (1999) for X a metrizable lcs, Y a Frechet space and ft cs-closed, while the statement and proof of Lemma 1.3.3 are those of Ursescu (1975); Corollary 1.3.4 (for Banach spaces) is due to Lifsic (1970). The statement of Theorem 1.3.5 is very close to that of [Kusraev and Kutateladze (1995), Th. 3.1.18]; when X, Y are Frechet spaces our statement is slightly more general. For X, Y metrizable locally convex spaces and CR. satisfying (Ha;) Theorem 1.3.5 is equivalent to the open mapping theorem of Simons (1990). Theorem 1.3.7 is obtained by Ursescu (1975) for Y0 := aff(Imft) = Y. It is established in [Robinson (1976)] for X, Y Banach spaces and Yo = Y; in this case the above result is met in the literature under the name of Robinson-Ursescu theorem. Corollary 1.3.8 was obtained by Amara and Ciligot-Travain (1999) for lcs-closed sets. Theorem 1.3.10 is due to Robinson (1976), while Theorem 1.3.11 can be found in [Li and Singer (1998)]. Theorems 1.3.16 and 1.3.17 are due to Carja (1989) (with different proofs). Section 1.4: The Ekeland variational principle [Ekeland (1974)] is met in the literature, generally, in the form given in Corollary 1.4.2; the statement and proof of Theorem 1.4.1 are from [Aubin and Frankowska (1990)]. The statement and proof of the Borwein-Preiss variational principle are from [Borwein and Preiss (1987)]. Theorems 1.4.5 and 1.4.6 can be found in the recent book [Carja (1998)]. Exercises: The exercises are generally known results or auxiliary results from various papers. So Exercise 1.1 is the famous Caratheodory theorem, Exercise 1.3 is from [Joly and Laurent (1971)], Exercise 1.5 is from [Laurent (1972)], Exercise 1.7 (for n = 2 and p = p) is from [Bauschke et al. (2000)], Exercise 1.10 is stated in [Li and Singer (1998)] in Banach spaces in a slightly weaker form, Exercise 1.12 is the main result of [Sullivan (1981)], Exercise 1.13 is from [Hamel (1994)], Exercise 1.14 is the known Caristi's fixed point theorem (see [Caristi (1976)]).

This page is intentionally left blank

Chapter 2

Convex Analysis in Locally Convex Spaces

2.1

Convex Functions

We begin this section by introducing some notions and notations. For a function / : X —¥ E and A g l consider: dom/

= {x e x | f(x) < +00},

e p i / = {(x,t) eX epi s / =

x E I f(x) < t},

{(x,t)\f(x)
[/ — 00 for every x £ X. It is evident that d o m / = Pry (epi/). Let X be a real linear space and / : X -> E; we say that / is convex if Var.l/GX, V A g [ 0 , l ] : /(Arc + (1 - \)y) < \f(x)

+ (1 - X)f(y),

(2.1)

with the conventions: (+oo) + (—00) = +00, 0-(+oo) = +00, 0-(—00) = 0. Note that when x = y, or A £ {0,1}, or {a;, y} (£. dom / , the inequality (2.1) is automatically satisfied. Therefore the function / is convex if V x , j / e d o m / , x ^ , V A G ] 0 , l [ : /(Aar + ( l - \ ) y ) < \f(x) + (1 - \ ) f ( y ) . (2.2) 39

Convex analysis in locally convex spaces

40

If relation (2.2) holds with " < " instead of " < " we say that / is strictly convex. Similarly, we say that / is (strictly) concave if —/ is (strictly) convex. Since every property of convex functions can be transposed easily to concave functions, in the sequel we consider, practically, only convex functions. To avoid multiplication with 0, taking into account the above remark, we shall limit ourselves to A G ]0,1[ in the sequel. In the following theorem we establish some characterizations of convex functions. Theorem 2.1.1

Let f : X —>• E. The following statements are equivalent:

(i) / is (strictly) convex; (ii) the functions tpxy : E —)• R, (px,y(t) := f((l — t)x+ty), convex for all x, y G X (x ^ y);

are (strictly)

(hi) dom f is a convex set and Vx,yedomf,

V A e ] 0 , l [ : f(\x + (1 - X)y) < Xf(x) + (1 -

X)f(y)

(the inequality being strict for x ^ y); (iv) V n € N, \/xi,...,xn f(XlXl

E X, VAi,...,A„ G]0,1[, A i + - - - + A„ = 1 :

+ • • • + Xnxn) < X1f{x1) + ••• + Xnf(xn)

(the inequality being strict when Xi,...,

(2.3)

xn are not all equal);

(v) epi f is a convex subset of X x E; (vi) epi s / is a convex subset of X x E. Proof. The implications (i) => (ii), (ii)=^(i), (i) =>• (hi), (iii) =4> (i) and (iv) => (i) are obvious. (i) =£• (v) Let {xi,ti),(x2,t2) 6 e p i / and A e]0,1[. Then x\,x2 G d o m / , f(xi) < ti and f(x2) < t2. Since / is convex, from Eq. (2.1) we obtain that f{Xxx + (1 - X)x2) < Xf{Xl)

+ (1 - X)f(x2)

< Xtx + (1 - X)t2,

whence X{xi,t\) + (1 - X)(x2,t2) € e p i / . Therefore e p i / is convex. (v) =^ (iv) Let k 6 N, A!,...,A fc <E]0,1[, AI + • • - + Xk = 1, xu ... ,xk e X. If there exists i such that f{x{) = oo, then Eq. (2.3) is obviously true. If f{xi) G E for every i, 1 < i < k, then (a;;, f{xi)) € epi / for every i; since e p i / is convex, J2i=i^i(xiif(xi)) € e Pi/> which proves that the inequality

Convex

functions

41

(2.3) is verified. Finally, suppose that /(x,) < oo for every i and that there exists io with f(xi0) = — oo; consider i; G E such that (x;,£i) 6 e p i / for every i ^ io', since ( i j 0 , —n) € e p i / for every n 6 N, we have /(AiXiH

hAfcXfc) < AitiH

hAj 0 _i£j 0 _i+Ai 0 (-n)+Ai 0 + ii i o + i+..

.+\ktk

for n G N. Letting n -»• oo in the above inequality we get f (J2i=i^ixi) = —oo; hence Eq. (2.3) is verified. The proofs for (i) => (vi) and (vi) => (iv) are similar to those of (i) =>• (v) and (v) => (iv), respectively. The proof for the "strictly convex" case is similar. • Note that in (v) and (vi) there are no counterparts corresponding to / strictly convex. Proposition 2.1.2 If f : X —> E is sublinear then f is convex. Moreover, f is sublinear if and only if e p i / is a convex cone with (0, —1) £ e p i / . Proof. Let / be sublinear. It is obvious that / is convex, and so e p i / is a convex set which contains (0,0). If (x,t) G e p i / and A > 0 then f(Xx) = \f(x) < Xt, and so X(x,t) G e p i / ; hence e p i / is a convex cone. Since /(0) = 0 > - 1 , we have that (0, - 1 ) g e p i / . Assume now that e p i / is a convex cone with ( 0 , - 1 ) ^ e p i / . It is immediate that / is convex (epi/ being convex) and f(Xx) = Xf(x) for A > 0 and x G X. So, for x,y E X we have that /(a; + y) = 2 / ( | x + | y ) < fix) + fiy)- I f /(0) < 0 then (0, -t) € e p i / for some t > 0, whence the contradiction (0, —1) 6 e p i / . Therefore /(0) = 0, and so / is sublinear. • The indicator function of the subset A of X is

iA:X->R,

iA(x) := j

0 +oo

if if

x G A, x e X\A.

Note that dom iA = A and epi t,A = A x R+. From the preceding theorem we obtain that LA is convex if and only if A is convex. If / is convex the sets [/ < A] and [/ < A] are convex for every A G E. The converse is generally false. A function / with the property that " [ / < A] is convex for every A G E" is said to be quasi-convex. So, / : X -» E is quasi-convex if and only if V z , j / G X , VAG[0,1] : / ( ( l - A)* + Ay) < max{/(a;),/(i/)}.

42

Convex analysis in locally convex spaces

The notion of convex function is extended naturally to mappings with values in ordered linear spaces. Let Y be a real linear space and Q C Y be a convex cone; Q generates an order relation on Y: y\ < 2/2 (or 2/1 • Y'; the operator H is Q-convex if Va;,2/eX, V A e ] 0 , l [ : H{Xx + (1 - X)y) < XH{x) + (1 -

X)H{y).

The operator H : X ->• F U {-00} is Q-concave if - i J : X -» F* is Q-convex. If A £ L(X, Y) then ^4 is Q-convex for every convex cone Q C Y. As in the case Y = E, the domain and the epigraph are defined by: domi? := {x £ X | #(2;) < 00} a n d e p i i ? := {(x, y) eX xY \ H(x) < y}. The characterizations of convex functions given in Theorem 2.1.1 are valid for a Q-convex operator. A function / : (Y, Q) —> E is Q-increasing if 2/1 f(,Vi) < /(2/2)• For such a function we consider that /(oo) = +00. It is obvious that every function is {0}-increasing, and that a linear functional ip : Y —> E is Q-increasing if and only if (p(y) > 0 for every y £ Q. One defines similarly the Q-decreasing functions. In the next result we mention some methods for deriving new convex functions from known ones. For a, (3 £ E we set a V /? := max{a,/?} and a A P := min{a, /3}. Theorem 2.1.3

Let X,Y

be linear spaces and Q CY

be a convex cone.

(i) / / fi : X —> E is convex for every i € / (/ ^ 0) then sup i g j fi is convex. Moreover, epi(sup i € / /,) = P | i 6 / e p i / j . (ii) / / / 1 , /2 : X -> E are convex and X £ E+, tften /1 + fi and A/i are convex, where 0 • /1 := tdom/i • Moreover, dom(/i -1-/2) = dom fi n dom fi,

dom(A/i) = dom / 1 .

(hi) If fn : X —t M. is convex for every n £ N and f : X —> E is such that f(x) = limsup n _ >0O fn(x) for every x £ X, then f is convex.

Convex

43

functions

(iv) If A C X xR is a convex set then the function ipA is convex, where ipA:X^R,


(v) / / F : X x Y —> E is convex, then the marginal ciated to F is convex, where h:Y^%

(2.4)

function

h asso-

h(y):=mix€XF(x,y).

(2.5)

(vi) Let g : Y —> E be a convex function. If H : X —> Y' is Q-convex and g is Q-increasing, then g o H is convex; this conclusion holds also if H : X —> Y U {—oo} is Q-concave and g is Q-decreasing. In particular, if A 6 L(X, Y) then go A is convex. (vii)

Let f : X —>• E, g :Y —> K 6e proper convex functions, and let

$,V:XxY->R,

{x,y):=f(x) + g(y),

¥(x,y)

:= f(x)V

g(y).

Then $ and \& are convex and proper. Moreover, d o m $ = d o m * = dom / x dom g, inf $ = inf / + inf g and inf $ = inf / V inf g. (viii) If f : X —> R is convex and A G L(X, Y) then the function Af:Y^R,

(Af)(y):=mi{f(x)\Ax

= y},

is convex. Moreover dom(Af) = A(domf) and inf Af = inf/. (ix) If / i , f2 : X —> R are convex and proper, their convolution max-convolution, defined by

and

f1Df2 : X -> K,

( / i D / 2 ) ( i ) := inf {/i(a;i) + / 2 (z 2 ) | an + x2 = x} ,

/1O/2 : X -> E,

(f10f2){x)

:= inf { / i ( n ) V / 2 (z 2 ) I Zi + *2 = z} ,

are convex. Moreover dom(/ 1 D/ 2 ) = dom^O./^) = dom/x + d o m / 2 , inf fiDfi = inf /1 + inf / 2 , inf f10f2 = inf /1 V inf f2, epi 4 (/iD/2) = epi s /1 + epi5 / 2

(2.6)

VA 6 E : [AO/2 < A] = [A < A] + [f2 < A].

(2.7)

and

Proof, (i) It is clear that epi(sup iejr fi) = f)ieIepifiSince fi is convex for every i, using Theorem 2.1.1, we have that epi/i is convex, and so epi(sup i € / fi) is convex. The conclusion follows using again Theorem 2.1.1. (ii) is immediate.

Convex analysis in locally convex spaces

44

(iii) Let A s]0,1[ and x,y € d o m / . Since limsup/„(x), limsup/„(y) < oo, there exists n0 £ N such that fn(x), fn(y) < oo for every n > no- Since / „ is convex, we have fn(Xx + (1 - X)y) < \fn(x)

+ (1 -

X)fn(y).

Taking the limit superior we obtain f(Xx + (1 - X)y) < Xf(x) + (1 -

X)f(y).

Therefore / is convex. (iv) Let (xi,ti), (x2,t2) £ episipA and A £]0,1[. Then there exist si,S2 £ ffi such that (xi,si),(x2,s2) € ^4 and si < £i, S2 < t2- Since A is convex, A(:ci,si) + (1 - A)(z2,S2) = {Xx\ + (1 - A)x2,Asi + (1 - A)s2) £ A. Therefore, (PA(XXI + (1 — A)x2) < Xsi + (1 — X)s2 < Aii + (1 — A)i2, and so X(xi,ti) + (1 — A)(a;2,t2) £ epi s y>A- Hence ip^ is convex. (v) Note that epish = P r y x R ( e p i s F ) ,

dom h = Pry (dom F).

(2.8)

The conclusion follows from Theorem 2.1.1(vi). (vi) We observe that (/ o H)(x) = inf y 6 y F(x,y), where F(x,y) := g(y) + t,ep\H(x,y), because / is (J-increasing. Since obviously F is convex, by (v) we obtain that (/ o H) is convex, too. The other case is proved similarly. The second part is immediate taking into account that A is {0}-convex and g is {0}-increasing. (vii) One obtains immediately that dom $ = dom $ = dom / x dom g and that $ and \? are convex. The formulas for inf $ and inf * are wellknown. (viii) We have that (Af){y) = mixZX F{x,y), where F(x,y) := f(x) + >-gTA(x,y)- It is obvious that F is convex. From (v) we obtain that Af is convex. As d o m F = {(x,Ax) \ x £ d o m / } , we have that dom(Af) — A(domf). The formula for inf Af is obvious. (ix) Let us consider the functions $, \P : X x X -y R defined by *(xux2)

:= fiixx) + f2(x2),

*(zi,x2)

:= h{xi)y

f2{x2),

and A : X x X -> X, A{x1,x2) := zi + x2. Then ( ^ D ^ ) ^ ) = (A$)(x) and (fiQf2)(x) = (A^!)(x). By (v) we have that $ and $ are convex; using (viii) we obtain that / i D / 2 and /1O/2 are convex, too. The relations on the

Convex

45

functions

domains, epigraph and level sets follow by easy verification. The formulas for inf / i D / 2 and inf /1O/2 follow immediately from (vii) and (viii). • The formulas (2.6) and (2.7) motivate the use of "epi-sum convolution" and "level-sum convolution" for the convolution and max-convolution, respectively. The convolution and max-convolution are extended in an obvious way to a finite number of functions; from Eqs. (2.6) and (2.7) one obtains that these operations are associative. The convex functions which take the value —00 are rather special. Proposition 2.1.4 Let f : X —»• E be a convex function. If there exists XQ € X such that f(xo) = —00 then f{x) = —00 for every x € ' ( d o m / ) . In particular, if f is sublinear and 0 S ' ( d o m / ) , then f is proper. Proof. Let x £ ' ( d o m / ) ; since xo £ d o m / , there exists /J, > 0 such that y := (1 + fi)x — \ix§ € d o m / . Taking A := (p, + 1 ) _ 1 e]0,1[, x = (1 — \)XQ + Ay, and so / O r ) < ( l - A ) / ( x 0 ) + A/(2,) = - o o . The case of / sublinear is immediate from the first part.

•

In the sequel we denote by A(X) the class of proper convex functions defined on X. Let now | / C c I and / : C -> M; we say that / is convex if C is convex and Vx,y€C,

V A € ] 0 , 1 [ : /(Ax + (1 - X)y) < Xf(x) + (1 -

\)f(y).

It is easy to see (Exercise!) that the function / above is convex if and only if

?-X^W

?M-(

f{x)

if

X e C

>

is convex in the sense of the definition given at the beginning of this section. Of course, we can proceed in the opposite direction; so a proper function / : X —>ffiis convex if and only if /|dom/ is convex in the above sense. The consideration of (convex) functions with values in M. has certain advantages, as we shall often see in the sequel. Taking into account the equivalence (i) <£> (ii) of Theorem 2.1.1, it is useful to know properties of convex functions of (only) one variable. The following result collects some properties of such convex functions.

46

Convex analysis in locally convex spaces

Theorem 2.1.5

Let f € A(E) be such that int(dom/) ^ 0.

(i) Let t\,ti € d o m / , t\ < £2. Suppose that there exists XQ €]0,1[ such that / ( ( l - A0)ii + X0t2) = (1 - A 0 )/(*i) + A 0 /(£ 2 )- Then VAe[0,l] : / ( ( l - A ) t 1 + A t 2 ) = (l-A)/(*i)+A/(*2), h-t s . t - t V t € [ t i , t 2 ] : /(*) = r—rf(ti) + i—±m). *t 2 2— - tt\ i

(2.9) (2.10)

12 — T\

(ii) Let to 6 d o m / . The function Vt0

: dom/ \ {i0} -»• K, <*„(*) == /( *1 f (*o), t — to

is nondecreasing; if f is strictly convex then tpt0 is increasing. (iii) Let to e d o m / . Tfte following limits exist: t4.to

I — to

*>to

I — to

r.(tt):-ta!StJM.mpmzmet, *t*o

t — to

t
(2.12)

t — to

and

/K*o)
(2.13)

moreover /l(£o),/+(*o) € K whenever t0 € int(dom/). Therefore f is left and right derivable at any point o/int(dom/). Moreover

r e [ / - W , / | ( t o ) ] n l « v t e i : T(t-t0) < f(t) - f{t0).

(2.14)

/ / / is strictly convex, in relation (2.14) the inequality is strict for t ^ to. (iv) Lei fi,i 2 £ d o m / , £1 < t 2 . Then f\.(h) < f'_{ti); if f is strictly convex then f\_{t\) < /!_(£ 2 ). Therefore the functions f'_ and f'+ are nondecreasing on d o m / . Furthermore, f is strictly convex if and only if f'_ [resp. f'+] is increasing on int(dom/). (v) The function f is Lipschitz on every compact interval included in int(dom/), and so f is continuous on int(dom/); moreover, for every to E int(dom/) we have: l i m / l ( i ) = l i m / ; ( i ) = f_(t0),

l i m / ; ( t ) = l i m / l ( 0 = f'+ifo).

tjlO

tllQ

tjtQ

t^-to

47

Convex functions

(vi) The function / is monotone on int(dom/), or there exists to & int(dom/) such that f is nonincreasing on ] — oo,£o] H d o m / and nondecreasing on [io,oo[n d o m / . (vii) Let to € d o m / and A 6]0,1[. Then the mapping tp : d o m / —> R, if>(t) := \f(t) — f(to + \(t — to)), is nonincreasing on Ii :=] — oo,to\Hdom/ and nondecreasing on Ir := [io,oo[ndom/. / / / is strictly convex then ip is decreasing on I\ and increasing on Ir. Proof. To begin with, let ti,t2,t3 € d o m / be such that t± < t2 < t3. Then t2 = {1 - X)tx + \t3 with A = (t3 - t2)/(t3 - h) € ]0,1[; therefore

m)<^-m) + ^ m ) ,

(2.15)

the inequality being strict if / is strictly convex. Subtracting successively f(h), f(h), f(t3) from both members of Eq. (2.15), multiplying then by and t£u> it,-tl)(tl-t2) A > respectively, we obtain

m)-m) t2 — t\

~

m)-w)

< /(f 3 )-/(* 2 ) )

t3 — tl t\ —t2 ~ f(tl) - /(* 3 ) < / ( * 2 ) - / ( * 3 ) > *i — *3

~

t3 —12 (216)

*2 — i 3

these inequalities being strict if / is strictly convex. (i) Let ti,t2 € d o m / and Ao G]0, l [ b e such that / ( ( l - A0)ii + A0i2) = (1 — A 0 )/(*i) + X0f(t2). Suppose that there exists A e]0,1[ such that / ( ( l - \)ti + Xt2) < (1 - X)f(h) + Xt2; assume that A < Ao (one proceeds similarly for A > A0). Taking 6 := (1 - Ao)/(l - A) e]0,1[, we have that A0 = dX + (1 - B) • 1 and (1 - A0)*i + A0*2 = 0((1 - A)*i + Xt2) + (1 - 6)t2; so we get the contradiction / ( ( l - X0)h + Xt2) < Of((l - X)h + Xt2) + (1 <6[(l-X)f(t1)

+ Xf(t2)} +

= (l-X0)f(t1)

+ Xof(t2).

9)f(t2) (l-6)f(t2)

Therefore Eq. (2.9) holds. Taking t€ [h,t2] and A = ( t - * i ) / ( i 2 - * i ) £ [0,1] in Eq. (2.9), we obtain Eq. (2.10). (ii) Let t 0 £ d o m / and tx,t2 £ d o m / \ {t0}, h < t2. Considering successively the cases t\ < t2 < to, h < t0 < t2, to < t\ < t2, from Eq. (2.16) we obtain that (ft0 is nondecreasing; if / is strictly convex, ipt0 is even increasing.

48

Convex analysis in locally convex spaces

(iii) To begin with, let to € int(dom/). Taking into account that ipt0 is nondecreasing on everyone of the nonempty sets d o m / n ] i 0 , oo[ and d o m / n ] — oo,t 0 [, the following limits exist: ,.hm(p (t),,x := hm ,. /(*) - /(to) • f /(«) ~ /(to) —- = inf -—- ^< oo, to

*4-*o

t\.to

,.

m

t — to

fit) ~ /(to)

r

t>to

1™ V«o(*) : = i m — : — r — =

SU

tjto

t
tfto

t — to

t — to

fjt) - f(t0) ^ P —:—:

> -°°>

t — to

and so /+(to) and / i ( t o ) exist. Since ipto is nondecreasing on dom / \ {to}, the inequality (2.13) holds. What was proved above shows that fL{to), f'+ito) G K. If to = max(dom/) then f(t) = oo for t > t0, whence /+(to) = oo; of course f'_ito) < oo (the existence of / i ( t 0 ) follows from the monotonicity of
-M

< f (to) < r < /;(to) <

im^lM.

t-t0 - • / - 1 - u ^ - - J +1 " ' t'-to Note that the first inequality, in the above relation, is strict if the first quantity is finite and the function / is strictly convex; similarly for the last inequality and the last quantity. From these inequalities we have immediately Eq. (2.14), with strict inequality if / is strictly convex. Conversely, assume that r 6 l and r(t —10) < fit) — /(to) for every t € K; dividing by t — to in each of the cases t > to and t < to and taking the limit for t —> to, we obtain that r 6 [/L(to),/+(to)](iv) Let ti,t2 £ d o m / , t\ < t2 and consider t €]ti,t 2 [; using Eqs. (2.11), (2.12) and (2.16) we obtain that

r+{h)

< /(')-/(«*> < m)-m) = /(«o-/(«a) < , ( }

the inequalities being strict if / is strictly convex. Using Eq. (2.13) we get that f'_ and f'+ are nondecreasing, even increasing if / is strictly convex. Suppose that / is not strictly convex; then there exists ti,t2 G d o m / , ti < t2, and A0 6]0,1[ such that / ( ( l - A0)ti + A0t2) = (1 - A 0 )/(ti) + A 0 /(t 2 )- Therefore Eq. (2.10) holds, whence

vte]tut2[-- f'-(t) = fW =

fit

?-{itl).

t2 — ti

Convex

functions

49

Because ]ii,i2[C int(dom/), we obtain that f'_ and f'+ are not increasing on int(dom/). (v) Let ti,t2 £ int(dom/), ti < t2 and t,t' e]ti,t2[, t < t'. From Eqs. (2.11), (2.12) and (2.16) we get

< /;(tl) J+K ; -

IMzM < m^M < t/('»)-/(«) <- -^^' r (ta)> ti-t f - t -t j

2

whence \f(t') - f(t)\ < M\t' - t \ , where M := max{|/^(*i)|, \fL(t2)\} G R Therefore / is Lipschitz on ]ti,<2[- Since every compact sub-interval of int(dom/) is contained in an interval ] i i , ^ [ with [ti,t2] C int(dom/), we get the desired conclusion. Let now to G d o m / be such that / is left-continuous at to (therefore to > inf(dom/)); for example to € int(dom/). We already know that V t e d o m / n ] - o o , t 0 [ : /!(*) < f+(t) < /l(*o) € ] - oo.oo]. Let A 6 E, A < / i ( i 0 ) ; from the definition of the least upper bound and relation (2.12), there exists *i 6 dom/, ii < to, such that A < (f(h) — f{to))l{t\ - t0). By the left-continuity of / at t0, there exists t2 £]ti,t0[ such that A < (/(ti) - f{h))/{h - t2). Let £ €]*2,*i[- Using again Eq. (2.16) we get A

fit) - f(t2) t-t2

=

f(t2) - f{t) t2-t

-J~{>-

Therefore \im^t0 f'-it) — u m tt (>)0. Indeed, if h,t2 E d o m / , t0 < h < t2, 0(<) < f'+ito) < f'+ih) < ifih) - fih))Ht2 - h). Similarly, if f_(t0) < (<)0 then / is (decreasing) nonincreasing on ] — oo, to] l~l dom / . So, from the above arguments, if /+(£) > 0 for every t £ int(dom / ) then / is nondecreasing on int(dom/); if /+(£) < 0 for every t G int(dom/), by Eq. (2.13), fL(t) < 0 for every t S int(dom/), whence / is nonincreasing on int(dom/). If none of these conditions is satisfied, there exist t\,t2 G int(dom/), h < t2, such that f'+(ti) < 0 < f'+(t2). Let t0 := inf{« € d o m / | f'+(t) > 0} €]h,t2]. It follows that f'_(t) < 0 for every t € dom / , t < t0, whence / is decreasing on ] - oo, t0] C\ dom / . By (v) we have that f'+{to) > 0, whence / is nondecreasing on [ t 0 , o o [ n d o m / .

50

Convex analysis in locally convex spaces

(vii) Let t i , t 2 G Ir be such that ti < t 2 . Then to + A(*i - t0) < h < t 2

and

t 0 + A(tx - t 0 ) < t 0 + A(t2 - i 0 ) < t 2 .

So, from the convexity of / , we have

m) < ^ffii^m

+ \(tl - to)) + t3%xl§;Xm),

f(to + X(h - to)) < ^ f f S T ^ / f a + A(ti - h)) +

I-^^L_f(t2),

the second inequality being strict if / is strictly convex. Multiplying the first inequality by A, then adding them, we get A/(*i) + /(to + A(t2 - t 0 )) < / ( t 0 + A(ti - t 0 )) + A/(t 2 ), i.e. V'(ti) < ipfo), the inequality being strict if / is strictly convex. The proof is similar on /(. • The preceding theorem shows that /|dom/ is continuous on int(dom/) when / : K -¥ K is a proper lower semicontinuous convex function. We have even more. Proposition 2.1.6 Let / G A(R). Then /| c i(dom/) JS upper semicontinuous (on cl(dom/)j. Moreover, if f is lower semicontinuous then /| c i(dom/) is continuous. Proof. As mentioned above, / is continuous at any to € int(dom/). Let to G d o m / . If d o m / = {to} it is nothing to prove. In the contrary case we can take to = inf(dom/) and some t € d o m / with t > to. Let (t„) C d o m / \ {to} converge to to. We may assume that tn
VfGE\dom/,

t
VtGK\dom/, t>sup(dom/).

Convex

functions

51

Theorem 2.1.7 Let ip : E —> E be a non-decreasing function and a G E fee SMC/I t/m£ V'C0) € E. Then /:E^E,

/(*):= J > ( s ) d s ,

is a proper lower semicontinuous convex function with I:— {s G R |
= f+,

(2-17)

where the integral is taken in Lebesgue sense and
ip+(t) := 'mi{ i},

for t G E. Moreover, if g : E —>-R is a proper lower semicontinuous function such that g'_ < ip < g'+ then g — f + a for some a G E.

convex

Proof. As usual, Ja ip(s) ds := — Jb ip(s) ds if a > b. The statement is obvious if / is a singleton. So we assume that i n t i ^ 0. If t G E \ c l / it is obvious that f(t) = oo. Hence I C d o m / C c l / . If
- /(*o) = / £ ¥>(*) ^ < (tA - t0)v(*A) = A(ti -

*O)V(*A),

f(ti) - f(tx) = Jtl (h - tx)
toMtx).

Multiplying the first relation by (1 - A) and the second one by -A, then adding them, we obtain that f(t\) < (1 — A)/(to) + A/(*i). Since f\cu is continuous we obtain this inequality also for to,ti G d o m / and A G]0,1[; hence / is convex. Let to G d o m / n c l / and t 6 E, t > to; because ip(s) > ip+(to) for s > to, we have that V(t) >

f{t) f{to)

-

t — t0

= - J - /%(s)cfe >
t o

and so / ' ( t 0 ) = ip+(t0). Similarly, fL(to) = ip-{to). Since these relations are obvious for t0 $ d o m / n c l / , Eq. (2.17) holds. Let now g : E —> E be a proper lsc convex function such that g'_ < ip < g'+. It follows that int(domg) = i n t / . Taking into account Theorem 2.1.5(v) we obtain that g'_(t) =
Convex analysis in locally convex spaces

52

and so g'_(t) = fL{t) and g'+(t) — /+(£) for every t € int I. Taking h : int I —> K, h(t) := f(t) — g(t), we have that h is derivable and h! = 0. It follows that ft is a constant function. Therefore there exists a G E such that g(t) = f(t)+a for all t € int I. Using the preceding proposition it follows that g = f + a. • The following consequence will be used several times. Corollary 2.1.8

Let f : E —• E be a proper convex function.

Then

\/t,t' e int(dom/) : /(*') - f(t) = / / ' /;(*) ds = / / ' / i ( s ) da. Moreover, if f is lower semicontinuous, then the preceding equalities hold for all t,t' € d o m / . Proof. Assume that int(dom/) ^ 0. When t := sup(dom/) e l o r t := inf(dom/) € E, we replace, if necessary, f(t) by limt.j.j/(£) and f(t) by lini£4.(/(£); then / is lower semicontinuous. Applying the preceding theorem for ip = f'+ and cp — f'_ the conclusion follows. D The following result is used frequently to show that a function of one variable is convex. T h e o r e m 2.1.9 Let I C E be a nonempty open interval and f : I —> E be a derivable function. The following statements are equivalent: (i) / is convex; (ii) V t , a e / : / ' ( * ) • ( * - * ) < / ( * ) - / ( * ) ; (iii) Vt,s € J : (f'{t) — f'(s)) (iv)

• (t — s) > 0, i.e. / ' is nondecreasing;

('i// is twice derivable on I) 'it £ I : /"(£) > 0.

Proof, (i) =$> (ii) Let / be convex. Since / is derivable we have that f'(s) = / i ( s ) = /+(s) for every s e I. The conclusion follows from Eq. (2.14) taking r = f'(s). (ii) =$• (iii) Let s,t £ I. By hypothesis we have that / ' ( * )

•

(

*

-

*

)

<

/ ( * )

-

/ ( * ) ,

/ ' ( * )

•

(

«

-

*

)

<

/ ( * )

-

/ ( * ) •

Adding these two inequalities side by side we obtain the conclusion.

Convex functions

(hi) =4> (i) Let tuh }tut2[. Then

53

€ I, h < t2, A €]0,1[ and tx := (1 - A)fi + Xt2 €

(1 - X)f(t1) + Xf{t2) = (l-\)[f(t1)-f(tx)]

f(tx) +

X(f(t2)-f(tx))

= (1 - AJ/'faXt! - t A ) + A/'(r 2 )(i 2 - tx) = A(l - A)/'(n)(ti - i 2 ) + A(l - A)/'(r 2 )(i 2 - tO = A(l-A)(i1-i2)(/'(r1)-/'(r2))>0, with Ti &]ti,t\[, r 2 €.]tx,t2[ (obtained by using the Lagrange theorem); of course, we have used the fact that / ' is nondecreasing. Suppose now that / is derivable of order 2 on J. In this case (iii) •& (iv) by a known consequence of the Lagrange theorem. • Theorem 2.1.10 Let I C R be an open interval and f : I —> R be a derivable function. The following statements are equivalent: (i) / is strictly convex; (ii) Vt,8el,t?8

: /'(*) • ( * - « ) < f(t) -

(iii) Vt,s £ I, t ^ s : (/'(*) — f'(s))

f(s);

• (t — s) > 0, i.e. f is increasing;

(iv) (if f is twice derivable on I) Vt E I : /"(*) > 0 and {t e I \ f"(t) — 0} does not contain any nontrivial interval. Proof. The proof is completely similar to that of the preceding theorem; just replace, where necessary, the inequalities by strict inequalities and, of course, use the properties of increasing functions. • Similar characterizations to those of the above theorems can be immediately formulated for concave and strictly concave functions. As immediate applications of the above two theorems we obtain that: (i) / i : R -» R, f\(t) := \t\p, where p £]l,oo[, is strictly convex; (ii) / 2 : R+ —> R, / 2 (£) := tp, where p e]0,1[, is strictly concave and increasing; (iii) / 3 : P -> R, f3(t) := tp, where p e R!_, is strictly convex and decreasing; (iv) fi : R -> R, fi(t) :- tp if t > 0, / 4 (t) := 0 if t < 0, where p e]l, oo[, is convex and nondecreasing; (v) /s : R -> R, /s(£) := exp(i), is strictly convex and increasing; (vi) / 6 : P -> R, / 6 (i) := lnt, is strictly concave and increasing; (vii) f7 : R+ -> R, f7(t) := tint if t > 0, /r(0) := 0, is strictly convex; (viii) f% : R ->• R, f%(t) := V l + 1 2 , is strictly convex. In the following two theorems we characterize the convexity of Gateaux differentiable functions defined on linear normed spaces.

Convex analysis in locally convex spaces

54

Theorem 2.1.11 Let D C (X, ||.||) be a nonempty, convex and open set, and / : £ ) - > E be a Gateaux differentiable function (on D). The following statements are equivalent: (i) / is convex; (ii) \/x,yED

:

(y-x,Vf(x))
(hi) Vx,y£D

: (y - x, Vf(y) - V/(x)> > 0;

(iv) Va; G £>, Vu G X : differentiable on D).

V2/(:E)(M,U)

> 0 (when f is twice Gateaux

Proof. For x,y G D consider Ix
+ ty).

(2.18)

It is well-known that ¥>*,»(*) = V/((l-t)a:+*y)(j/-a:),

<

W = V2

f{{l-t)x+ty)(y-x,y-x) (2.19) for every t G / x ^ ; of course, the second formula is valid when / is twice Gateaux differentiable on D. (i) => (ii) Let x,y € D. By Theorem 2.1.1 we have that (iii) Writing the hypothesis for the couples (x,y) and (y,x) we obtain immediately the conclusion. (iii) => (i) Let x,y € D and t, s G 7x>y, s < t; then y

0<(t-s)(
s)x - sy, V / ( ( l - t)x + ty) - V / ( ( l - s)x + sy)).

Using Theorem 2.1.9 we obtain that ipX:y is convex, whence, by Theorem 2.1.1, / is convex. Suppose now that / is twice Gateaux differentiable on D. (i) =*> (iv) Let x € D and u € X. Taking into account that D is open, D - x is absorbing; hence there exists a > 0 such that y := x + au G D. From the hypothesis we have that ipx>y is convex, whence by Theorem 2.1.9, < Px,y(t) > 0 for every t G Ix>y. In particular < „ ( 0 ) = V 2 /(*)(t/ -x,y-x)=a2

V2f(x)(u,u)

> 0.

Convex

55

functions

Therefore the conclusion holds. (iv) => (i) Let x,y € D. For every t G Ix,y, using Eq. (2.19), we have 0. Applying again Theorem 2.1.9 we obtain that ipx,y is convex, and so / is convex. • In the next result we give characterizations for strictly convex functions. Theorem 2.1.12 Let D C (X, \\ • ||) be a nonempty, convex and open set, and f : D -» E be Gateaux differentiable (on D). Then (i) <$• (ii) •£> (hi) •£= (iv), where (i) / is strictly convex; (ii) Vx,yeD,x?y

: (y - x, V/(z)) < f{y) -

(iii) Vx,y£D,x^y

f(x);

: (y - x, V/(y) - V/(x)) > 0;

(iv) \/x E D, V w £ l \ {0} : S72f(x)(u,u) Gateaux differentiable on D).

> 0 (when f is twice

Proof. Of course, using Theorem 2.1.1, we know that / is strictly convex if and only if the function y given by Eq. (2.18) is strictly convex for all x,y G D, i / j / . The proof is similar to that of the preceding theorem, using Theorem 2.1.10 instead of Theorem 2.1.9. • A remarkable property of convex functions is given below. Theorem 2.1.13 there exists

Let f G A(X) and x G d o m / . Then for every u e X

f(x, u) := lim ^ J v ' no

+

fa)

-^}

t

= inf & t>o

+ tu)

' ™ t

G I,

(2.20) '

K

and VueX

: f'(x,u)
+ u)-f(x),

(2.21)

the inequality being strict if f is strictly convex, u ^ 0 and f'(x,u) < oo. Moreover fix, •) is sublinear and dom f'(x, •) = cone(dom/ — x). If x G 4 (dom/) then f'(x,-) is proper, while if x £ (dom/) 1 then f'(x,u) G K for every u € X. Proof. Let u £ X and rp : R -> R, tp(t) := f(x+tu). Taking into account that VJ — ifiXtX+u (
Convex analysis in locally convex spaces

56

2.1.5 we obtain the existence of

^(Q)=lim^-f^inf^-f), i.e. Eq. (2.20) holds. Using again Theorem 2.1.5 we obtain Eq. (2.21), with the corresponding variant for / strictly convex and u ^ 0. It is obvious that f'{x,0) — 0 and that f'(x,Xu) = Xf'(x,u) for every u G X and A > 0. Let now u,v £ X. We have f(x + t(u + v)) = f ( | ( x + 2tu) + \{x + 2tv)) < \f(x + 2tu) + \f{x + 2tv), and so f(x + t(u + v)) - f(x) f(x + 2tu)-f(x) t ~ 2t

f(x +

2tv)-f(x) 2t '

Letting t ]. 0 we obtain V«,«£l

: f'(x,u

+ v) < f'(x,u)

+

f'(x,v).

Therefore f'(x, •) is sublinear. Note that dom/'(x, •) = E+ • ( d o m / - x) = cone(dom/ — a;). If a; € *(dom/) then dom f'(x,-) = lin(dom/ — x), whence 0 € l (domf'(x, •)). From Proposition 2.1.4 we have that f'(x, •) is proper. If a; € (dom /)* then dom f'(x, •) = X and f'(x, •) is proper, which proves that f'(x,u) G M for every u £ X. • The number f'(x, u) G 11 is called the directional derivative of / at x in the direction u. It is possible that f'(x,-) take the value — oo. Consider the function / : R -> E, /(*) := - ^ 1 - i 2 if |i| < 1, / ( i ) := oo if |i| > 1; we have /'(—l,i) = - o o for every t > 0 (Exercise!). The preceding result can be extended in a significant way. Theorem 2.1.14 function

Let / G A(X), x G d o m / anrf e G M+. Then the

fe{x,) : * - > ! ,

^,«):=jnf / ( a ; +

to) /(3;)+e , t-

is sublinear, dom /^(x, •) = cone(dom / - x),

(2.22)

Convex functions

57

and ft(x,u)
+ u)-f(x)+£

= l i m fe{x,u)

V«6l,

= inf fe{x,u)

Furthermore, if x £ *(dom/) £/ien f'e{x,-) i/ien f'e{x,u) € E /or every u £ l .

(2.23)

Vw £ X.

is proper, while if x £ ( d o m / ) 1

Proof. From the definition of f'e{x, •) it is clear that f'E(x,u) < oo if and only if there exists t > 0 such that x + tu 6 dom f, i.e. u £ cone(dom f — x). Hence Eq. (2.22) holds. It is obvious that f^(x,0) = 0 and for A > 0 fs{x,Xu)

= inf

—

A=

Xfe(x,u).

Let now u,v e X and s,£ > 0. Then / (x + ^ ( « + « ) ) = / ( ^ ( s + *u) + ^ ( z + tvj) <^rtf(x

+ su) + ^rtf(x

+ tv).

It follows that (f(x+£i(u

+

v))-f(x)+s)/£ +t f(x + su)- f{x)+e f(x + < ~ s

tv)-f(x)+e t

Therefore, for all s, t > 0 we have /(a! + g u ) - / ( g ) + g , / ( a +

fa)-/(aQ+e

Taking the infimum in the right-hand side, successively with respect to s and t, we obtain that £ ( x , u + i ; ) < / ; ( z , u ) + £(£,
< f'ei(x,u)

<

f'S2(x,u)

Convex analysis in locally convex spaces

58

hold, we obtain x i-limf. tu i . u \ = inf • f f'(x,u) tit \ = •mif- inf + f f(

= inf inf / ( * + * " ) - / ( * ) + £ t>0£>0 t = f'(x,u) for every u £ X. theorem.

tu

x £ )-f( - )+

f(x + t u ) ~ t>0 t

= inf

f(x)

The other conclusions are proved like in the preceding •

The number f'e{x,u) € K is called the £-directional derivative of / at x in the direction u. We end this section with a characterization of convex functions in arbitrary topological vector spaces using a generalized directional derivative. We shall give other characterizations in Section 3.2 using generalized subdifferentials. So, let X be a topological vector space and / : X -> M a proper function. We define the upper Dini directional derivative of / at x 6 X in the direction « £ l b y Df(x

u\

_ / limsupt4.0t_1(/(a;-l-tu)-/(x)) \ — oo

if

x € dom/, otherwise.

When / is convex and x G d o m / from Theorem 2.1.13 we have that Df(x,u) = f'(x,u) for all x € d o m / and u € X. For the proof of the announced characterization we need the following auxiliary result which is interesting in itself, too. Lemma 2.1.15 Let a, b 6 M, a < b, and (p : [a,b] —>• E be a lower semicontinuous proper function with 0. Proof. By contradiction, assume that Dtp(t, 1) < 0 for every t 6 [a, b[. Fix a > 0; we have that tp(t) <
\
Because ip is lsc, it follows that ip(t) < ip(a) + a(t — a). Assume that t < b. Then, as above, there exists t" E]t,b] such that
Convex functions

59

ip(a) + a(i — a) + a(t — i)= ip(a) + a(t — o) for t £ [M"[, contradicting the choice of t. Hence t = b, and so desired inequality is proved. As the inequality tp(b) < tp(a) + a(b — a) holds for all a > 0, we obtain the contradiction ip(b) < f(a). The conclusion holds. • Theorem 2.1.16 Let X be a topological vector space and f : X ->• R be a proper lower semicontinuous function. Then f is convex if and only if Df{x,y-x)+Df(y,x-y)<0 for all x, y £ X for which the sum makes sense. Proof. Assume first that / is convex. If x £ d o m / then Df(x,y — x) = —oo, and so Df(x, y — x)-\- Df(y, x — y) equals — oo or does not make sense. Assume that x,y £ d o m / . Then Df(x,y-

x) = f'(x,y-

Df(y, x-y)

x) < f(y) -

f(x),

= f(y, x - y) < f(x) -

f(y);

summing the inequalities side by side we get the conclusion. We prove the sufficiency by contradiction. Assume that / is not convex. Then there exists x,y € d o m / , x ^ y, and A g]0,1[ such that / ( ( l - X)x + \y) > (1 - A)/(z) + Xf(y). Let

I be defined by (*) := / (tx + (1 - t)y) - tf(x) - (1 -

tf(y), t)f(y).

Of course,

(1 - A) = ifi(X), and D
- t)x + ty,y-x)

+ f(x) - f{y)

^V(*. 1) = Df(tx + (1 - t)y, x - y) + /(y) - f(x)

V t e [0, A[, Vt 6 [0,1 - A[.

Prom the preceding lemma we get t\ € [0, A[ and ti £ [0,1 — A[ such that ~Dtp(ti,l) > 0 and 'Dip{t2,1) > 0. Taking u := (1 - ii)z + ^ y and v := ^2^+(1 — ^2)2/, we have that u — v = s(x — y) with s := 1 —ti—*2 S]0,1]. Moreover, Df(u, v — u) + Df(v, u - v) = s-1 (p
= 8- (D
+ Dil>(t2,l)) > 0 ,

f(y))

Convex analysis in locally convex spaces

60

a contradiction. The proof is complete.

2.2

•

Semi-Continuity of Convex Functions

In this section X is a separated locally convex space if not stated explicitly otherwise. We begin with some characterizations of lower semicontinuous convex functions. Theorem 2.2.1

Let f : X —> R. The following conditions are equivalent:

(i) / is convex and lower semicontinuous; (ii) / is convex and w-lower

semicontinuous;

(iii) epi / is convex and closed; (iv) epi / is convex and w-closed. Proof. It is well-known that: / is lsc •£> e p i / is closed in I x I « [/ < -M is closed VA £ 1. The equivalence of conditions (i)-(iv) follows immediately applying Theorem 2.1.1. D In the sequel we shall denote by T(X) the class of proper lower semicontinuous convex functions on X. The following criterion for convexity is sometimes useful. Theorem 2.2.2 Let f : X —> K be a proper function satisfying the following conditions: (i) /(0) = 0, (ii) V i £ l , VA 6 P : /(Ax) = Xf(x), (iii) / is quasi-convex. Suppose that either (a) or (b) holds, where (a) V i £ X : f(x) > 0, (b) d o m / C cl{a; e X | f(x) < 0} and f is lsc at every point x G d o m / . Then f is sublinear. Proof. Note first that f(x + y) < f{x) + f(y) if x,y E d o m / and either f(x) • f{y) > 0 or 0 = f{x) < f(y). Indeed, if f{x) • f{y) > 0 then

whence f(x + y) < f(x) + f(y). If 0 = f(x) < f(y), for n £ N we have that J£T/(*

+ V) = f(nTTnx

+ n^iV) < max{f(nx),f(y)}

= f(x) + f(y).

Taking the limit for n —• oo we obtain again that f{x + y)
Semi-continuity

of convex

functions

61

Suppose that (b) holds. Let A := {x G X \ f(x) < 0} C d o m / . If x € d o m / \ ^ 4 , by hypothesis, there exists a net (a;i)iej C A, with (xi) —>• a;. Moreover, 0 < f(x) < \iminfieif(xi)

< limswpieIf(xi)

< 0,

whence (/(a;,)) —> /(a;) = 0. Therefore f(x) < 0 for every x € d o m / . Let x,y E d o m / . Since the roles of x,y are symmetric, we have to consider the following three situations: a) x,y £ A, ft) x,y G d o m / \ A and 7) x G d o m / \ A and 2/ G A. In the case a) we have that f(x)-f(y) > 0, and so f(x + y) < f(x) + f(y). In the case /3) we have that f(x) = /(y) = 0 and a; + y G d o m / , whence f(x + y) < 0 = / ( x ) + / ( y ) . In the case 7) there exists a net (xi)iei C A such that (xi) —> a;; then (/(a:;)) —>• 0. Since a; + y G d o m / and /(a^ + y) < f(xi) + f(y) for every i e J [by a)], taking the limit inferior we obtain that f(x + y) < f(x) + f(y). • Corollary 2.2.3 Let f : X —> R be a quasi-convex, proper, positively homogeneous and Isc function, / + := / V 0 and g := f + LC\A, where A := {a; G X I f{x) < 0} U {0}. Then f+ and g are sublinear and f = / + A g. Proof. It is obvious that / + and g are quasi-convex, lsc and positively homogeneous. The subhnearity of /+ and g follows applying the preceding theorem. Because f(x) < 0 for x G cl-A, we have also that / — /+ A g. • A result of the same type is given by the following corollary. Corollary 2.2.4 Let f : X —> E be a quasi-convex and positively homogeneous function. If f is upper bounded on a neighborhood of the origin or dimX < 00, then f+ is sublinear. Proof. Suppose that there exists Ao > 0 such that [/ < Ao] is a neighborhood of 0. Since [/ < A] = ^ [ / < A0] for every A > 0, we have 0 G int[/ < A] for every A > 0. Let 0 < A < fi and x G cl[/ < A]. Since 0 G int[/ < A], from Theorem 1.1.2 we have that ^x G int[/ < A]. Hence x G f int[/ < A] = int (f [/ < A]) = int[/ < p] C [/ < »]. Therefore VA > 0 : cl[/ < A] C f |

\f
Hence [/ < A] is a closed set for every A > 0. Since [/ < 0] = HM>o[^ — lA> the set [/ < 0] is closed, too. Since [/+ < A] = [/ < A] for A > 0 and

Convex analysis in locally convex spaces

62

[f+ < A] — 0 for every A < 0, it follows that / + is lsc. Taking into account that / + is quasi-convex, /+ is sublinear (applying Theorem 2.2.2). Suppose now that dimX < oo. By hypothesis, [/ < 1] is convex and absorbing. Since dimX < oo, as observed at the end of Section 1.1, we obtain that 0 G int[/ < 1]. The conclusion follows as above. • Note that, under the conditions of the preceding theorem, /+ is continuous (see Theorems 2.2.9 and 2.2.21). A result similar to that of Proposition 2.1.4 is the following. Proposition 2.2.5 Let f : X —» E be a lsc convex function. If there exists XQ G X such that f(xo) — — oo, then f(x) = —oo for every x G dom / . In particular, if f is sublinear then f is proper. Proof. Suppose that there exists x G d o m / such that f(x) =: t G E. Then (x, t) G epi / and (xo,t — n) G epi / for every n G N. It follows that V n G N : ±(x0,t - n) + (1 - ±){x,t) = (±x0 + *^x,t

- l) G e p i / ;

therefore (x,t — 1) G cl(epi/) = e p i / , whence the contradiction f(x) < f{x) - 1. • Propositions 2.1.4 and 2.2.5 motivate the consideration, in the sequel, of (almost only) proper convex functions. It is useful to consider the lower semicontinuous envelope or lower semicontinuous regularization / := ci(ePi/) of the function / : X —> E (see Eq. (2.4)). Because cl(epi/) is closed we have that e p i / = cl(epi/). Theorem 2.2.6

Let f : X —> K be a convex function.

Then

(i) / is convex; (ii)

if g : X —> E is convex, lsc and g < f then g < f;

(iii) the function f does not take the value —oo if and only if f is bounded from below by a continuous affine function; (iv) if there exists xo G X such that f(xo) = — oo (in particular if f(xo) = — ooj then f(x) = —oo for every x G d o m / D d o m / . Proof, (i) Since e p i / = cl(epi/) and e p i / is convex, we have that e p i / is convex; hence / is convex. (ii) Let g be lsc, convex and g < / ; then e p i / C epig. Taking the closure of the sets, the conclusion follows.

Semi-continuity

of convex

functions

63

(iii) Suppose that / does not take the value — oo. If f(x) = oo for every x € X then f(x) > (x, 0) + 0 for every x € X. Consider dom / ^ 0 and let x € dom / . Then (x, t) fi epi / , where t := f(x) — 1. Since epi / is a convex, closed and nonempty set, using Theorem 1.1.5 we obtain (x*,a) € X* x E such that V (a;, t) € epi / : {x,x*) + at < (x, x*) + at. Taking x = x and t = f(x) +n, n £ N, we obtain that a < 0. Dividing, by —a > 0, we can suppose that a = — 1. Thus Vz e d o m / D d o m / :

f{x)>f(x)>(x,x*)+j,

where •y :—t— {x,x*}. Therefore / is bounded from below by a continuous affine function. Conversely, if f(x) > (x,x*) + 7 =: g(x), where x* G X* and 7 £ R, then g is convex and lsc; by (ii) we have that g < f. Therefore / does not take the value —00. (iv) If f(xo) = - 0 0 , using the preceding theorem, f(x) — —00 for every x 6 d o m / . It is obvious that d o m / D d o m / . • Proposition 2.2.7 Assume that f : X —> ffi is sublinear. sublinear o- / is lsc at 0 •£> / is proper.

Then f is

Proof. Because e p i / = cl(epi/) is a convex cone, the first equivalence follows by Proposition 2.1.2. If / is sublinear, by Proposition 2.2.5 we have that / is proper. Conversely, if / is proper then 0 > /(0) > —00 which implies that 7(0) = 0. • To an arbitrary function / : X —> E one associates, naturally, a lower semicontinuous and convex function; this function is denoted by co/ and has the property that epi(co/) = cl(co(epi /)) and is called the lsc convex hull. It is obvious that co/ < / < / . An application of the lsc convex hull of a function is the property in the next example used in the study of Hamilton-Jacobi equations. Example 2.2.1 Let X be a locally convex space, / € r ( X ) , g : X -> E be a proper function and a, 0 > 0. Then co(co(f+ag)+/3g) =co(f+(a+/3)g). (See the solution of Exercise 2.19 for details.) Related to the upper semi-continuity of a convex function, we remark that when / : X -> E is upper semicontinuous (use for short) at x0 € dom / ,

64

Convex analysis in locally convex spaces

f is upper bounded (by a real constant) on a neighborhood of xo; therefore xo G int(dom/) 7^ 0. In the sequel we prove that the converse is true for every convex function. Let us first establish the following auxiliary result. Lemma 2.2.8 Let f G A(X) and x0 G d o m / . Suppose there exist U G J4CX and m Gffi_|_such that VxGx0

+ U : f(x) < f(x0) + m.

(2.24)

Then Vxexo + U : \f(x)-f(x0)\<mpu(x-x0),

(2.25)

where pi; is the Minkowski gauge ofU. In particular f is continuous atXQ. Proof. Replacing / by g : X ->• R, g(x) = f(x0 + x) — f{x0), we may suppose that xo = 0 and f(xo) = 0. Therefore f(x) < m for every x € U. Prom Proposition 1.1.1 we have that pu is a continuous semi-norm and U = {x G X I pu{x) < 1}. Let x 6 U. Suppose first that t :— pu{x) > 0. Then y :— t~1x € U. Since / is convex and t < 1, we obtain that f(x) < tf{y) + (1 - t)f(0) < tm = mpu{x). Suppose now that pu{x) = 0. Then nx G U for every n € N. It follows that f(x) < ±f(nx) + ^ / ( O ) < \m for every n 6 N. Therefore, once again, f(x) < mpu(x). Since 0 = / (\x' + \{-x')) < \f{x') + y(-x'), we obtain that -f(x') < f(-x') for every x' e X. Using this inequality we obtain that \f(x)\ < mpu(x) for every x eU. Therefore Eq. (2.25) holds. • Using the preceding lemma we get the following important result. Theorem 2.2.9 / / the convex function f : X —> K is bounded above on a neighborhood of a point of its domain then f is continuous on the interior of its domain. Moreover, if f is not proper then f is identically - c o on int(dom/). Proof. Suppose that there exist Xo £ d o m / , V G N(xo) and m G M such that f(x) < m for every x G V. Then V C d o m / , and so xo G int(dom / ) 7^ 0. If / takes the value - 0 0 then, by Proposition 2.1.4, f(x) = —00 for every x G int(dom/). Therefore the conclusion holds in this case. Let / be proper and x G int(dom/). Then there exists /J, > 0 such that xi := (1 + fi)x - nx0 G dom / . Taking A := (1 + p,)~l G ]0,1[, we have that / (An + (1 - X)x) < A/(xi) + (1 - \)f(x)

< \f(Xl)

+ (1 - A)m =

: m i

6R

Semi-continuity

of convex

functions

65

for every x € V. But V\ := Xxi + (1 — A)V £ N(x). Applying the preceding lemma we obtain that / is continuous at x. • Corollary 2.2.10 Let f : X —>• E be a convex function. Then f is continuous on int(dom/) if and only i/int(epi/) is nonempty in X x E. Proof. Suppose that / is continuous at some XQ £ int(dom/). Then, for m &]f(xo),oo[, there exists U £ N x such that f(x) < m for every x e xo + U. So (xo, m + 1) G int(epi/) because (xo + U) x [m, oo[ C e p i / . Conversely, assume that (xo,M £ hit(epi/); then there exist U £ N x and e > 0 such that (xo + U) x [t0 — e, t0 + e[ C epi / , and so f(x) < m := t0 + e for every x 6 xo + U. By Theorem 2.2.9 we have that / is continuous on int(dom/) ( ^ 0). • Under the conditions of Lemma 2.2.8 we have a stronger conclusion. Theorem 2.2.11 Let f € A(X) and XQ £ d o m / . Suppose that there exist U € J^cx and m 6 E+ suc/i t/iat condition (2.24) is satisfied. Then for every p e]0,1[, Vx,y £x0+pU

: |/(cc) - f(y)\ < mpu(x-y). 1-p Proof. As in the proof of Lemma 2.2.8, we may assume that xo = 0 and /(a;o) = 0. Let p e]0,1[ and consider x,y 6 pU. We consider two cases: a) pu{x — y) ^ 0 and b) pu{x — y) = 0. In the proof we shall use the fact that pu is a continuous semi-norm, U = {x \ pu{x) < 1} and intf/ = {x | pu{x) < 1} (see Proposition 1.1.1). a) Let

tpu{x — y) — pu(—y), and so limt_,.oo ¥>(*) = oo. As (p(l) = pu(x) < P < 1, there exists t > 1 such that ¥>(£) = 1. Then 2 := te+(l-t)j/ £ U. It follows that x = (l-i~1)y + i-1z, and so f(x) < (1 — i~1)f(y) + i~1f(z). From Lemma 2.2.8 we have that \f(y) I < mpu(y) < wp. Therefore /(*) - f(v) < ^

(m

- f(y)) < * _ 1 (m + mp).

But i{x-y) - z-y, whence ipu(x-y) - pu(z-y) >Pu(z)-pu(y) > 1-p. The preceding inequalities imply that fix) - f(y) < m\^-Pu{x 1-p

- y).

(2.26)

66

Convex analysis in locally convex spaces

b) In this case we have, using again Lemma 2.2.8, that f(t(x — y)) = 0 for every t £ i So, WG]0,1[

:

f(tx)=f(ty+(l-t)J^(x-y))
Since x G int U C int(dom / ) , by Theorem 2.2.9, / is continuous at x. From the above inequality we obtain that f(x) < f(y) taking the limit for t —> 1. Hence (2.26) also holds in this case. The conclusion follows from Eq. (2.26) changing x and y. • Corollary 2.2.12 Let (X, ||-||) be a normed space and f G A(X). Suppose that XQ G dom / and for some p > 0 and m > 0, Vz G D(x0,p)

: f(x) < f(x0) + m.

Then Vp'e]0,p[,Vx,yeD(xQ,p')

: \f(x) - f(y)\ < - • P-^ • \\x - y\\. p p-p'

Proof. Consider U := D(0;p) — pUx- Then pu{x) = p _ 1 ||x|| for any x 6 X. The conclusion is immediate from the preceding theorem. • The conclusion of Theorem 2.2.11 says, in fact, that / is Lipschitz on a neighborhood of xo • We say that the function / : X —> K is Lipschitz on a set A C X if / is finite on A and there exists a continuous semi-norm p on X such that \f(x) — f(y)\ < p(x — y) for all x,y E A; we say that / is locally Lipschitz on A if for every x G A there exists a neighborhood V of a: such that / is Lipschitz on V. Corollary 2.2.13 If f £ A(X) is bounded above on a neighborhood of a point of its domain then f is locally Lipschitz on the interior of its domain. Proof. Let x G int(dom/). Applying Theorem 2.2.9 we obtain that / is continuous at x, and so / is bounded above on a neighborhood of a;. Applying now Theorem 2.2.11 we obtain that / is Lipschitz on a neighborhood of a;. • Recall that in Theorem 2.1.5 we have already proved that a function / G A(E) is locally Lipschitz on int (dom/), while in Proposition 2.1.6 we proved that /|dom/ is continuous if / is, moreover, lsc. Another consequence of Theorem 2.2.9 is the following result.

Semi-continuity

of convex

functions

67

Corollary 2.2.14 Let ft 6 A{X) forl
(- dom / „

and either f (resp. g) is identically —oo on int(dom/) (resp. int(dom<7),); or f (resp. g) is proper and continuous on int(dom/) (resp. int(domg)^.D Recall that ftOfa and /1O/2 are defined in Theorem 2.1.3. Prom Theorem 2.2.9 (or Corollary 2.2.10) we obtain that g is continuous on int(domp) if f,g are convex, g < f and / is continuous on int(dom) (supposed to be nonempty). A similar result is true for a larger class of convex functions. The convex function / : X —• E is said to be quasi-continuous if aff(dom/) is closed and has finite codimension (i.e. its parallel linear subspace has finite codimension), rint(dom/) 7^ 0 and /| a ff(dom/) ls continuous on rint(dom/). The set A C X is quasi-continuous if LA is quasicontinuous; it follows that A is quasi-continuous exactly when aff A is a closed affine set of finite codimension and rint^4 7^ 0. The following result holds. Proposition 2.2.15 Let f,g : X —>• E be convex functions such that g < f. If f is quasi-continuous, then g is quasi-continuous, too. Proof. Without loss of generality we assume that 0 S dom / and /(0) < 0. Then aff (dom / ) is a linear subspace and Y0 := aff (epi / ) = aff (dom / ) x E. Of course Y0 has finite codimension and is closed. Moreover, by Corollary 2.2.10, we have that inty 0 (epi/) 7^ 0. Since e p i / C epig, we have that inty 0 (y 0 nepip) ^ 0. It follows (see Exercise 1.3) that rint(epi5) 7^ 0. Since aff (epi g) = aff (dom g) x E, using again Corollary 2.2.10, we obtain that g |aflf(domp) i s continuous, and so g is quasi-continuous. • Applying the preceding result to indicator functions one obtains Corollary 2.2.16 B.

Let A C B C X. If A is quasi-continuous then so is D

There are several classes of convex functions, larger than the class of lower semicontinuous convex functions, which will reveal themselves to be useful in the sequel. We introduce them now. Let / : X —> E. We say that / is cs-convex if

E,

n

.* , ™

Afc/(xfc) fc=l

68

Convex analysis in locally convex spaces

whenever J2n>i ^nxn is a convex series with elements of X and sum x € X. Of course, if / is cs-convex then / is convex, while if / is lsc and convex, / is cs-convex (Exercise!). Also, we say that / is ideally convex, bcscomplete, cs-closed, cs-complete, li-convex or lcs-closed if epi / is ideally convex, bcs-complete, cs-closed, cs-complete, li-convex or lcs-closed, respectively. Of course, taking into account the relationships among these notions for sets and Proposition 2.2.17 (i) below, we have: / lsc, convex

=>•

/ sc-convex

af cs-complete =>• / cs-closed =>• f lcs-closed ^ ^ ^ / bcs-complete =>• f ideally convex => / li-convex

=> f convex,

the reversed implications being not true, in general. Taking into account Proposition 1.2.3 and that for / : X -» R we have [/ < A] = Fix (epi/ n (Xx } - oo, A]), [ / < A] = P r x ( e p i / n ( X x ] - co, A[), the sets [/ < A] and [/ < A] are li-convex (lcs-closed) for every A € R if / is li-convex (lcs-closed). Let A C X; since epi LA = A X R+ and ffi is a Frechet space, we have that A is ideally convex (bcs-complete, cs-closed, cs-complete, li-convex, lcs-closed) if and only if i& is so. Remark 2.2.1 When

R is a continuous affine functional (that is

R is cs-convex (ideally convex, cs-closed) if and only if / + is so (Exercise!). Proposition 2.2.17

Let f, g : X —>• R have nonempty

domains.

(i) / / / is cs-convex then f is cs-closed. Conversely, if f is cs-closed and is minorized by a continuous affine functional then f is cs-convex. (ii) / / / and g are ideally convex (resp. cs-closed) and are minorized by continuous affine functionals then f + g is ideally convex (resp. cs-closed). Moreover, if g is bcs-complete (resp. cs-complete) then f + g is bcs-complete (resp. cs-complete). Proof. Taking into account the Remark 2.2.1 above, when / or/and g is bounded from below by a continuous affine function we may assume that even / > 0 or/and g > 0.

Semi-continuity

of convex

functions

69

(i) The proof of the first part is immediate. Suppose that / is cs-closed and / > 0. Let X) n >i ^nXn be a convergent convex series with elements of X and sum x G X. Since / is convex (epi/ being convex), we may suppose that An > 0 for every n £ N; moreover, we may assume that xn G dom / for every n. Because f(xn) > 0, there exists r := l i m n - ^ X)*!=i ^kfi^k) G EU{oo}. If T < oo, because (xn,f(xn)) € e p i / and J2n>i ^n{xn,f{xn)) = (X,T), and / is cs-closed, we have that (x,r) £ e p i / , whence f(x) < r. If T = oo, the preceding inequality is obvious. Hence / is cs-convex. (ii) We prove only the second part of the "ideally convex" case, the rest of the proof being similar. So, let / be ideally convex, g be bcs-complete and / , g > 0. Let J2n>i ^n(xn,rn) be a Cauchy b-convex series with elements of epi(/ + g). Then rn = sn + tn with (xn,sn) G e p i / n and (xn,tn) G epig for every n. Because 0 < sn,tn < rn, we have that (sn), (tn) are bounded, s := ^2n>i ^nSn € 1 + , t := 5 2 n > 1 Xntn G M+ and r = s + t. Because g is bcs-complete we obtain that the b-convex series ^2n>1Xn(xn,tn) with elements of epi g is convergent with sum (x,t) £ epig for some x G X. Hence the b-convex series Yln>i ^n{xn,sn) with elements of e p i / is convergent with sum (x, s) G e p i / . Therefore (x,r) G epi(/ + g). Thus f + g is bcs-complete. • The classes of li-convex and lcs-closed functions have good stability properties. Let us begin with the following characterizations. Proposition 2.2.18 Let / : X -» K have nonempty domain. Then the following statements are equivalent: (i) / is li-convex (resp. lcs-closed); (ii) epi s / is li-convex (resp. lcs-closed); (iii) there exist a Frechet space Y and an ideally convex (resp. cs-closed) function F : X x Y -» E such that f(x) = inf yC y F(x,y) for every x G X (i.e. f is the marginal function associated to an ideally convex (resp. csclosed) function). Proof. We prove the "li-convex" case, the proof for the other case being similar. (i) => (ii) Taking % § : X =} E such that gr 31 = epi / and gr S = X x P, we have that "R and S are li-convex multifunctions. Since E is a Frechet space, using Proposition 1.2.5 (iv), epi s / — e p i / + {0} x P = gr(IR + §) is li-convex.

70

Convex analysis in locally convex spaces

(ii) =$• (i) Since epi,, / is li-convex, as above, the set An := epi s / + {0}x ] — i,oo[ is li-convex for every n 6 N. Since e p i / = f]n&NAn, from Proposition 1.2.4 (i) we obtain that e p i / is li-convex. (i) => (iii) Since / is li-convex, there exist a Frechet space Y and an ideally convex set A C Y x X x E such that e p i / = PIXXR(A). Consider the function F: XxYxR-^R defined by F(x,y,r) := r + iA(y,x,r). Then f(x) = inf ( a . i r ) e e p i / r = i n f ^ ^ ^ r = inf ( j, ) r ) e y x R F(a;,j/,r) for every x £ X. Since A is ideally convex, so is LA] using Proposition 2.2.17 (ii) and Remark 2.2.1 we obtain that F is ideally convex. The conclusion follows because Y x E is a Frechet space. (iii) =>• (ii) Let f{x) = inf y e y F(x, y) for every a; € X, where Y is a Frechet space and F is an ideally convex function. From (i)=^(ii) it follows that epi s F is li-convex. Then, by Eq. (2.8), epi s / is li-convex. • Other useful properties of li-convex and lcs-closed functions are collected in the following result. Proposition 2.2.19 (i) / / / „ : X -> E is li-convex (resp. lcs-closed) for every n E N , then s u p n e N / „ is li-convex (resp. lcs-closed). (ii) If fi,h '• X -» E are li-convex (resp. lcs-closed) functions and A € E+, then / i + / 2 and A/i are li-convex (resp. lcs-closed). (iii) If F : X xY ^ R is li-convex (resp. lcs-closed) and X is a Frechet space, then h : Y —»• R, h(y) := inixex F(x,y), is li-convex (resp. lcsclosed). (iv) Let Y be a Frechet space and g : Y —> R. If Q C Y is a convex cone, H : X —>• (Y',Q) has li-convex (resp. lcs-closed) epigraph and g is li-convex (resp. lcs-closed) and Q-increasing, then g o H is li-convex (resp. lcs-closed). In particular, if A : X —> Y is a linear operator with li-convex (resp. lcs-closed) graph and g is li-convex (resp. lcs-closed), then g o A is li-convex (resp. lcs-closed). (v) / / X is a Frechet space, A : X —> Y is a linear operator with liconvex (resp. lcs-closed) graph and f : X —>• E is li-convex (resp. lcs-closed), then Af is li-convex (resp. lcs-closed). (vi) If X is a Frechet space and / i , / 2 : X -> E are li-convex (resp. lcs-closed) functions then / i D / 2 and /1O/2 are li-convex (resp. lcs-closed).

Semi-continuity

of convex

functions

71

Proof. Again, we treat only the "li-convex" case. (i) Because epi (sup n € N /„) = ri n eN e Pi/™' * n e conclusion follows using Proposition 1.2.4 (i). (ii) Taking % : X =t R, g r # i := epi/j (i = 1,2), we have that epi(/i + / 2 ) = gr(3?i +3?2)- The conclusion follows from Proposition 1.2.5 (iv). Also, epi(A/i) = T(epi/i) for A > 0, where T : X x E - > X x R i s the isomorphism of topological vector spaces given by T(x, t) = (x, Xt); hence A/i is li-convex in this case. If A = 0, A/i = tdom/i- But d o m / i = Prjt(epi/i), and so d o m / i is li-convex by Proposition 1.2.3, whence 0/i is li-convex. (hi) We have, by Eq. (2.8), that epi s / = Pry X R(epi s .F). Since episF is li-convex and X is a Frechet space, we get from Proposition 1.2.3 that epi s / is li-convex. (iv)-(vi) Using the same constructions as in the proofs of Theorem 2.1.3 (vi), (viii) and (ix), the conclusions follows from (iii). • If X is a barreled space, the lower semi-continuity of a convex function (even weaker conditions) ensures its continuity on the interior of its domain. Theorem 2.2.20 Let X be a barreled space and f : X —> E be convex. Suppose that either (a) X is first countable and f is li-convex or (b) / is lower semicontinuous. Then ( d o m / ) ' = int(dom/) and f is continuous on int(dom/). Proof, (a) By Proposition 2.2.18, there exist a Frechet space Y and a li-convex function F : X xY -> R such that f(x) = inf y € y F(x, y) for every x £ X. Consider the multifunction X : 7 x l r j I with gr3? := {(y,t,x) \ (x,y,t) G epiF}. Of course, Ji is ideally convex and ImR = d o m / . Let xo € (dom f)1 (if this set is nonempty) and {ya,tQ) € 3l _1 (a;o). Applying Simons' theorem (Theorem 1.3.5), we have that U := R(Yx ] - 00, t0 + 1[) is a neighborhood of x0. So, for every x € U there exist y 6 Y and t < to + 1 =: m such that f(x) < F(x,y)
72

Convex analysis in locally convex spaces

Proof. By Proposition 1.2.1 we have that e p i / is cs-closed, and so / is lcs-closed. The conclusion follows from the assertion (a) of the preceding theorem. • The application of Corollary 2.2.12 and Theorem 2.2.20 yields the following uniform boundedness principle for convex functions. Theorem 2.2.22 Let X be a Banach space and C C X be an open convex set. Consider {fi)i^i o, nonempty family of continuous convex functions from C into E. If (fi{x)).j is bounded for every x £ C then for every x £ C there exist rx,Lx > 0 such that Ux := x + rxllx C C and \fi(y) — fi(z)\ < Aclly - z\\ for all y,z £ Ux and all i £ I (i.e. (fi) is locally equi-Lipschitz on C). Proof. By hypothesis, for every x £ C there exists Mx £ E+ such that \fi(x)\ < Mx for all i £ J. Let r'x > 0 be such that U'x := x + r'xUx C C and consider Fi : X -> E be denned by Fi{y) := fi(y) for y £ U'x and Fi(y) := oo otherwise. Then Ft £ T(X). Consider F := supieI Ft E T(X). The hypothesis shows that d o m F = Ux. By Theorem 2.2.20 we obtain that F is continuous on int(dom.F) — x + rxBx- Therefore there exist r 'x £]°><[ and M > 0 such that F(y) < M for all y G x + r'J,Ux. Then for y G x + r'J.Ux and i G / we obtain that ft(y) - fi(x) < M — (-Mx) = M+Mx =: L'x. Using now Corollary 2.2.12 we have that fi is Lipschitz with constant Lx :~ ZL'x/r'^ o i i x + rxUx, where rx := r'^/2. The conclusion follows. • Taking X a Banach space, Y a normed space, and {Xi | i G 1} C £ ( X , Y) (I 7^ 0) with {TiX \ i £ 1} bounded for every x £ X, the conditions of the preceding theorem are satisfied by the family of functions (fi)iei, where U{x) := ||Ti(a:)||, and C := X. Hence \\Ti(x)\\ = \h(x) - / 4 (0)| < L||x|| for x £ rllx and i £ I, for some r,L > 0; hence \\Ti\\ < L for every i £ I. So we obtained the classic uniform boundedness principle in Functional Analysis. From the preceding result we obtain that pointwise convergence implies uniform convergence on compact sets. Corollary 2.2.23 Let X be a Banach space and C C X be an open convex set. Consider (/„) a sequence of continuous convex functions defined on C with values in E such that (fn(x)) -> f(x) for every x £ C with f : C —> E. Then (/„) converges uniformly to f on the compact subsets ofC

Semi-continuity

(or, equivalently, (fn{xn)) to x E C).

of convex

73

functions

-> f(x) for every sequence (xn) C C converging

Proof. First of all observe that / is convex (by Theorem 2.1.3) and locally Lipschitz (by the preceding theorem). Let x,xn E C (n E N) be such that (x„) -> x. By Theorem 2.2.22 there exist r, L > 0 such that Ux := x + rUx C C and \fn{y) - fn(z)\ x, xn E £/x for n > n x for some nx E N. Then |/n(*n) - f(x)\ < \fn(xn) <M\\xn-x\\

- fn(x)\ +

+ \fn(x) - f(x)\ \fn(x)-f(x)\

for n > nx, whence (fn(xn))n -*• f(x). Assume now that there exists some compact subset K of U such that (/„) does not converge uniformly to / on K. Then there exist e > 0, P C N an infinite set and a sequence (xn)nep C K such that \fn(xn) — f(xn)\ > £• Since K is compact, we may assume that (xn) -> x 6 K...C C. Then, as shown above, {fn{xn))p -> f{x). Since / is continuous, we have also that (f(xn)) • p —>• f(x), which yields the contradiction 0 > e. The next result corresponds to a known theorem for continuous linear operators. Proposition 2.2.24 Let X be a Banach space and C C X be an open convex set. Assume that / , / „ : C —> M. (n € N) are continuous convex functions. Then (fn(x)) —> f(x) for every x 6 C if and only if (a) (fn{x)) is bounded for every x G C and (b) (fn(x)) -> f{x) for every x € D for some dense subset D of C. Proof. The necessity is obvious. Assume that conditions (a) and (b) above are satisfied but (fn{x)) does not converge to f(x) for some x E C. Hence there exist e > 0 and P C N an infinite subset such that \fn(x) - f(x)\ > e for every n E P. By Theorem 2.2.22 we find r,L > 0 such that Ux := x + rUx C U and / , / „ (n E N) are L-Lipschitz on Ux. Since D is dense, there exists (xk) C D converging to x; we assume that Xk E Ux for every fcgl Then for n E P and k E N we have S < \fn(x)

- f(x)\

< \fn(x)

< 2L \\xk - x\\ + \fn{xk)

- fn(Xk)\

-

f{xk)\.

+ \fn(xk)

~ f(xk)\

+ \f(xk)

-

f(x)\

Convex analysis in locally convex spaces

74

Fixing k G N such that \\xk - x\\ < e/(4L), we get |/„(x fc ) - f(xk)\ for n G P, contradicting that (fn(xk) - n G N) converges to f(xk).

> e/2 •

Let / G T(X); we call the recession function of / the function /oo : X —>• IK whose epigraph is ( e p i / ) ^ . Let XQ G d o m / ; taking into account formula (1.3), we have that (u, A) G epi/oo < » V i > 0 : (x 0 ,/(x 0 )) + £(u, A) G e p i /

/(x 0 + fa) - /(x 0 ) ^

<

Jim

/(X 0 + fa) - / ( x 0 )

t-S-OO

£

= gup

/(Xp + fa) -

(>0

/(XQ) < A

t

~

Therefore VtiSl

: /oo(u) = lim t—>oo

7

;

(2.27)

r

thus /oo(u) > - o o for every u G X and /oo(0) = 0. Because epi/oo is a closed convex cone we have that f^ is a lsc sublinear functional. The preceding relations show that f(x + u) < /(x) + /oo(u)

V x G d o m / , Vu G X,

[/ < A]^ = {u e X | /«,(«) < 0} V A G E w i t h [ / < A ] # 0 ,

(2.28) (2.29)

when / € T(X). Taking into account the discussion on page 6, relation (2.29) shows that the function / € T(X) has bounded level sets when dimX < oo. This result is no longer true when dimX = oo; take f.i. / := PA, the Minkowski functional associated to the set in Example 1.1.1 (/oo = / in this case). Remark 2.2.2 Note that, for / G F(X) and x G X, the mapping t >->• / ( x + fa) is nonincreasing from E into E when /oo(u) < 0; in particular d o m / + E+u = d o m / . Moreover, if /<»(«) < 0 and /oo(~w) < 0 then f(x + tu) = f{x) for all x G X and t G E. Indeed, let ti < t2. If / ( x + t\u) < oo then / ( x + t2u) - f(x + tiu) _ f(x + tin + (t2 - h)u) - f(x + tiu) t2 —t\ t2 — *i

Conjugate

75

functions

and so f(x + t2u) < f(x + t\u); the inequality is obvious if f(x + tiu) = oo. It follows that d o m / + R+u = d o m / . When /«>(«) < 0 and /oo(-w) < 0 we can apply the previous result for u and — u and obtain that the mapping t H-> f(x + tu) is constant.

2.3

Conjugate Functions

In this section X and Y are separated locally convex spaces if it is not stated explicitly otherwise. Let / : X —> M; the function f*:Xm-+%

f*{x*):=suv{(x,x*)-f(x)\xeX},

(2.30)

is called the conjugate or Fenchel conjugate of / . Note that if there exists XQ 6 X such that /(xo) = ~°o then f*(x*) — oo for every x* € X*, while if f(x) = oo for every x then f*{x*) = - o o for every a;* £ X*. When / is proper (but also for / not proper, using the convention inf 0 = oo), we have f*(x*)

= sup{(x,a;*) - f(x) | x € d o m / } .

The conjugate of a function h : X* -> K. is defined similarly: /i* : X - > K ,

/i*(a;) := sup{(a;,a;*) - h{x*) \ x* e X*}.

In fact, considering the natural duality between X and X*, it is the same definition; this distinction is useful in the case of normed spaces. The above remark concerning /* is also valid for h*. In the following theorem we establish several simple properties of conjugate functions. Theorem 2.3.1

Let f,g

: X -> I , h : Y ->• 1 , k : X* -)• I and

A€l(X,Y). (i) /* is convex and w*-lsc, k* is convex and Isc; (ii)

the Young-Fenchel

inequality

V x e l . V i ' e r

below holds:

: f{x) +

f*(x*)>(x,x*);

(iii) f<9^9*
76

Convex analysis in locally convex spaces

(v) if a > 0 then (af)*(x*) = af'ia^x*) for every x* G X*; if/3 ^ 0 tfien (f(l3-))*(x*) = f'ifi-^x*) for every x* G X*; (vi) if g(x) = f(x + x0) for x G X, then g*(x*) = f*(x*) every x* G X*;

- {x0,x*)

for

(vii) if x*0 G X* then (/ + x*0)*{x*) = f*(x* - x*0) for every x* G X*; (viii) if f, h are proper and $ : I x 7 -> 1, $(x, y) := f(x) + h(y), then $*(x*,y*) = f*(x*) + h*{y*) for all (i*,y*) E X* x Y*; (ix)

{AfY

=f*oA*

and (fDg)* = f* + g*.

Proof, (i) If / is not proper, we have already seen that / * is constant, and so / * is convex and w*-continuous. If / is proper we have that / * = su Pzedom/ *Px, where ipx : X* -*• R, ipx(x*) := {x,x*) - f(x). It is obvious that for every x G d o m / , (px is affine (hence convex!) and u>*-continuous (hence w*-lsc!). Therefore / * is convex and u;*-lsc. For the statement about h we use the same arguments. (ii) By Eq. (2.30) we have VxeX,Vx*

eX*

: /'(a;*)>
which gives immediately the Young-Fenchel inequality. (iii) is an immediate consequence of the definition and of the relation f<9_ _ (iv) We already remarked that co/ < f < f, whence, using (iii), we get / * < / * < (co/)*- Let x* G X* and a G R be such that f*(x*) < a. Then (x,x*) — f(x) < a for every x G X, whence ip(x) := (x,x*) — a < f(x) for every x. Since tp G T(X), and epi(p D e p i / , we have that epiy> D epi(co/) = co(epi/). Therefore (f(x) < (cof)(x) for every x G X, and so (x,x*) — cof(x) < a for every x; hence (co/)*(a;*) < a. Thus / * = / = (co/)*(v), (vi), (vii) and (viii) are immediate. (ix) We have (Afy(y*)

= sup{(y,y*) - (Af)(y)] = sup ((j,,**) inf f(x)) y£Y y£Y V {x\Ax=y} J = sup{(y,y*) - f{x) \(x,y)£X =

x Y, Ax = y}

sup{(x,A*y*)-f(x)\x£X}

= f*(A*y*) =

(roA*)(y*).

Similarly one proves the corresponding relation for (fDg)*.

D

Conjugate

77

functions

I the sequel we denote by T*(X*) the class of those functions in A(X*) which are w*-lower semicontinuous. The discussion at the beginning of this section and the preceding theorem yields the next result. Corollary 2.3.2

Let f : X -> E and h : X* - • E. Then

(i) /* € T*(X*) <S> d o m / ^ 0 and 3x* G X*, a G K, Vrr G X : f(x) > (x,x*) + a ; (ii) h* G r ( X ) <S> dom/i / 8 and 3 i 6 I , a e E, Va:* £ X* : /i(:r*) > (x,a;*) + a. • The following result is fundamental in duality theory. Theorem 2.3.3 and f **:=(/*)*

Proof.

(of the biconjugate) Let f e T(X).

Then f* € T*{X*)

= f.

Applying Theorem 2.2.6 we get x$ G X* and a Gffisuch that V i e l

: / ( i ) > (x,x*0)+a.

(2.31)

Thus, by the preceding corollary, we have that /* G T*(X*). Moreover, Theorem 2.3.1 shows that /** < / . Let x € X be fixed and consider t G E such that t < f(x); therefore (x, t) ^ e p i / . Applying Theorem 1.1.5 for {(x,t)} and e p i / , there exist (x*,a) G X* x E and A G E such that V ( z , i ) G e p i / : (x,x*) + ta < A < (x,x*) + ta.

(2.32)

Taking (x, t) = (x, f(x) + n), n G N, with x G d o m / , we obtain that V n G N : (x,x*) + af(x) + na < A < (x,x*) + ta. Letting n -> oo we obtain that a < 0. Take first a < 0. Dividing eventually by —a > 0, in Eq. (2.32) we can suppose that a = —1. Thus (x,x*) - f(x) < A V i £ d o m / . Hence /*(#*) < A < (x,af) - <, and so

t<(x,x*)-r(r) 0 such that V z G d o m / : {x,a;*) + c < (x, x*).

78

Convex analysis in locally convex spaces

This together with Eq. (2.31) yields f(x) > (x,

XQ)

+ a > (x,

XQ)

+ a + t(x, x*) + tc — t(x, x*)

for all x € dom / and all t > 0; this implies successively: MxeX,

V i > 0 : -tc + t(x,x*) - a> (x,XQ+tx*) V £ > 0 : -tc + t{x,x*) -a

V i > 0 : f**(x)

-

f(x),

> f*(x*Q + tx*),

> (x,x*0+tx*)-f*(x*0+tx*)

> a+

tc+(x,x*0).

But there exists t > 0 such that a + tc + (x, a;*,) > t; hence f**(x) > t in this case, too. Thus we obtained f**(x) > f{x). Therefore /** = / . • The preceding theorem shows that for any function / : I -> 1 we have ((/*)*) = /* (for the duality (X,X')) which shows that there is no interest to consider conjugates of order greater than 2. It also shows that the conjugation is an isomorphism between T(X) and T*(X*). More precise information on the biconjugate of an arbitrary function is furnished by the following result. Theorem 2.3.4 (i) —oo.

Let f : X —>•ffihave nonempty domain.

/ / co/ is proper, then /** = c o / ; if co/ is not proper, then /** =

(ii) Suppose that f is convex. If f is Isc atxG d o m / , then f(x) /**(x); moreover, if f(x) € R, then /** = / and f is proper.

=

Proof, (i) The function co/ is convex and lsc. If co/ is proper, using the preceding theorem and Theorem 2.3.1(iv), we have that

co/= (co/r = (/•)* = /**• If co/ is not proper, since dom(co/) D dom / / 0, co/ takes the value - c o ; hence /* = (co/)* = oo, and so /** = —oo. (ii) Taking into account that / is convex, co/ = / . Since / is lsc at x, we have that f(x) = f(x). It is obvious that f**(x) = f(x) if f(x) = - o o . Let f(x) £ M; then f(x) € E, and so / is proper. From the first part we have that /** = J, whence f**(x) = J(x) = f(x). • Corollary 2.3.5 Let f,g £ A(X). If fUg is proper, then (/* + g*)* = fOg = fOg~, while in the contrary case (/* +
: (fDg)(x)

=

(f*+g*)*(x).

The sub differential of a convex

Proof.

function

79

From Theorem 2.3.1 we have that

uogr = r+g* = T+r = (f^9)*\ since /•
sA(x*) :=sup{(x,x*)\x

&A};

(2.33)

for 0 ^ B C X* the support function SB : X —> E is defined similarly. It is obvious that SA is w*-lsc and sublinear while SB is lsc and sublinear. Furthermore, if C C X is another nonempty set then SA+C — $A + $c and SAUC — SA V sc- Note that (IA)*(X*)

= sup(a;,a;*) = sA(x*) = sup (x,x*) = x£A

(LCOA)*(X*)

= ScoA-

X€COJ4

Moreover we have that icoA — cotyi and dom SA = dom(tJ4)* = {x* € X* | x* is upper bounded on A}; therefore doms^ = X* if and only if A is u;-bounded. In the next section we shall see that LA is useful in determining the normal cone of C. 2.4

The Subdifferential of a Convex Function

We have already seen in Section 2.1 that if the proper convex function / : (X, || • ||) -> M. is Gateaux differentiable a t i G int(dom / ) then

V z e d o m / ( V a ; e X ) : (x -x, Vf(x)) < f(x) - f(x). Taking into account this relation, it is quite natural to consider the elements x* E X* which satisfy the inequality VxEX

: (x-x,x*)
even if / is not Gateaux differentiable at x.

(2.34)

80

Convex analysis in locally convex spaces

In this section the spaces under consideration are separated locally convex spaces if not stated otherwise. Let / : X -¥ E and x £ X be such that f(x) G E. An element x* G X* is called a subgradient of the function / at x if relation (2.34) is satisfied; the set of all the subgradients of the function / at x is denoted by df(x) and is called the subdifferential or Fenchel sub differential of / at x. We consider that df (x) = 0 if f(x~) £ E; of course we can have df(x) = 0 even if f(x) G E. Thus we obtain a multifunction df : X =4 X*. By the preceding considerations we have that domdf C d o m / . We say that / is subdifferentiable at x G X if df(x) ^ 0. Note that if x* G df(x), the afflne function tp : X —• R, (p(x) := (x,x*) — (x,x*) + f(x) minorizes / and coincides with / at x; this proves that V ( x , i ) € e p i / : (x,x*) — t < a := (x,x") — f(x), which shows that the hyperplane {(x,t) € X x R \ (x, x*) — t • 1 = a] is a, non vertical (since the coefficient of t is ^ 0) supporting hyperplane (see p. 5 for the definition). Recall that in Theorem 2.1.5 we have already determined the subdifferential of a function / 6 A(M) at to £ dom / :

0/(*o) = [/:(*o),/;(to)]nR. In the sequel we shall establish properties of the subdifferential and methods for computing it also for X ^ E. In the next result we collect several easy properties of the subdifferential. Theorem 2.4.1

Let f : X -> 1 and x 6 X be such that f(x) 6 E. Then:

(i) df(x) C X* is a convex and w*-closed (eventually empty) set. (ii)

Ifdf(x)^
and

d(cof)(x)

= dj(x) =

df(x);

in particular /** = co/ and f is proper and Isc at ~x. (iii) / / / is proper, dom f is a convex set and f is subdifferentiable at every x £ dom / , then f is convex. Proof,

(i) Let x\, x*2 e df{x) and A G ]0,1[. Then

V z G X : (x-x,x*1)
(x-x,x*2)
The sub differential of a convex function

81

Multiplying the first relation with A > 0, the second with 1 - A > 0, then adding them, we obtain

Vx£l :

(x-x,Xx*1+(l-X)x*2)
whence Ax* + (1 — \)x% G df(x). Let x* G X* \ df(x). Then there exists XQ G X such that (xo — x,x*) > /(xo) - fix). Let a G E be such that {x0 — x,x*) > a > f(x0) — f{x). Then V := {x* \ (x0 -x,x*) > a} is a neighborhood of x* for the topology w* = a(X*,X). It is obvious that V n df(x) - 0. So df(x) is w*-closed. (ii) We already know that co/ < f < f- Let x* G df(x) and y>:K->K,

ip(x):=(x-x,x*)

+ f(x);

(fi is convex, continuous and

: (x-x,x*)

< (cof){x) - (co/)(z) < f{x) -

f(x),

whence d(cof)(x) C df(x). Therefore d(cof)(x) = df(x) = df(x). (iii) By (ii) we have that f(x) = cof(x) for every x G d o m / . Since co/ is a convex function and dom / is convex, it is obvious that / is convex. • The property (ii) of Theorem 2.4.1 justifies why we consider only proper convex functions when discussing subdifferentiability (in the sense of convex analysis!). In a similar way we introduce the subdifferential of a function h : X* —> l a t a point x* G X* with h{x*) G E: dh(x") = {x € X | Vx* G X* : {x,x* - x*} < h(x*) -

h(x*)}.

(As for conjugation, this distinction is important only when working with normed spaces; in the case of locally convex spaces we always consider the natural duality between X and X*, when the dual of X* is X.) Similar results to those of Theorem 2.4.1 hold; more precisely dh(x*) is a closed convex (eventually empty) set and if dh(x*) ^ 0 then h is proper and w*-\sc at x*; the other statements are also valid if one takes the closures with respect to the weak* topology.

82

Convex analysis in locally convex spaces

In practice (for example for solving numerically problems using computers) it is possible to determine the subgradients only approximately. In this sense the following notion of subgradient reveals itself to be useful. Let / : X ->• 1 , x G X with f(x) G M. and e G ! + ; the element x* G X* is called an e-subgradient, of the function / at x~ if Vxel

: (x-x,x*)


(2.35)

the set of e-subgradients of / at x is denoted by def(x) and is called the e-subdifferential of / at ~x. As for the subdifferential, if f(x) ^ M we consider that def(x) = 0; we obtain a multifunction dEf : X =£ X* with dom(<9E/) C d o m / . Note that / is proper if d£f{x) ^ 0 for some e > 0; if 0 < £i < £2 < oo then Of(x) = d0f(x)

C 3 ei /(af) C 5 e a / ( i ) .

Moreover VeGl^

: 9e/(i) = fl

a,/(i).

The £-subdifferential of a function /t: I * -> K at ? € I * with /i(x*) G E is introduced similarly. In the following theorem we collect several properties of the subdifferential and of the £-subdifferential. Before stating this theorem, let us introduce some notions: we say that the set M C X x X* is monotone if V(x,x*),(y,y*)€M

: (a: - y,x* - y*) > 0;

M is strictly monotone if V{x,x*),

(y,y*)eM,

x^y

: (x - y,x* - y*) > 0;

M is maximal monotone if a) M is monotone and b) M' C X x X* monotone and M C M' imply M = M', i.e. if M is a maximal element of the class of monotone subsets ordered by inclusion. We say that the multifunction T : X =3 X* is monotone, strictly monotone or maximal monotone if gr T is monotone, strictly monotone or maximal monotone, respectively. Of course, when X = E and T is single-valued (i.e. T(a;) is a singleton for every x G domT), T is (strictly) monotone if and only if the function T|d om T is nondecreasing (increasing). In the next result we do not assume that the functions are convex.

The subdifferential

of a convex

83

function

Theorem 2.4.2 Let f,g : X ->• R, h : Y -> R be proper functions, A 6 L(X,Y), x G domf n domg, y G dom/i and e E M+. T/»en: (i) dsf(x)

is a convex w* -closed set.

(ii) x* G 3 e / ( z ) «• /(x) + f*(x*)

<(x,x*)+e^x£

def*(x*).

(iii) a;* G df(af) «• /(x) + /*(a;*) < (x,x*) & f(x) + f*(x*) xedf*(x*). (iv) dom(<9£/) C d o m / , lm(dsf)

= (x,x*) =>

C dom/* and df is monotone.

(v) 3/(3;) ± 0 «• /(3f) = m a x a . e x . ((x,x*) -

f*{x*)).

(vi) de(f + x*)(x) = x* + def(x) for x* E X*; 0e(A/)(3:) = \ds/xf(x) and d(Xf)(x) — Xdf(x) for X > 0; if g(x) = f(x + XQ) for x G X then dEg(x) = dEf(x + x0). (vii) Assume that y = Ax; then A* (deh(y)) C de(h o A)(x). ( vii i)

U^6[0,e] (dvf(*) + 9S-Vg(x))

C 3 £ ( / + g)(x).

(ix)

Suppose X is a normed space, f is Isc, £j > 0 and (xi,x*) G

grdEif

for every t £ 7. If (ei)ig/ —> e < oo and either (a) (xi)iei —• a;,

(x*)iei -^-> a;* and (x*)ie/ is norm-bounded or (b) (x»)i e / -^-> £, (a;j)j e j is norm-bounded and (x*) —t x*, then (x,x*) G grdsf. In particular grdef is closed in X x X* (for the norm topology). Proof, (i) The fact that dEf(x) is convex and u>*-closed is shown similarly to the first part of Theorem 2.4.1. (ii) We have that x* ed£f(x)

oVxGX

: (x - x,x*) < f{x) - f(x)

» V i £ l

: {x,x*)-f(x)<(x,x*)-f(x)

+e +e

&f*(x*)<(x,x*)-f{x)+e &f{x)

+

Assume now that x* G def(x). f"(x)

f*(x*)<(x,x*)+e. Then, by what precedes,

+ /*(**) < f(x) + fix*)

<

(x,x*)+e,

and so x € d£f*(x*). (iii) We already know that (37, x*) < f(x) + f*(x*) (the Young-Fenchel inequality!); the equivalences follow now from (ii) taking e = 0. (iv) It is obvious that dom(<9£/) C d o m / , while the inclusion Imdef C d o m / * follows from (ii).

84

Convex analysis in locally convex spaces

Let (x,x*), (y,y*) £ gr<9/. Then x,y £ d o m / and {y - x, x*) < f(y) - f(x),

(x - y,y*) < f(x) - f(y).

Adding these relations we get (y — x,x* — y*) < 0, and so df is monotone. (v) If x* € df(x), taking into account (iii) and the Young-Fenchel inequality, we have Vx'el*

:

(x,x*)-r(x*)=f(x)>(x,x*)-r(x*),

and so the implication "=^" is true. The converse implication is an immediate consequence of the equivalences of (iii). (vi)-(viii) follow easily from the definition of the e-subdifferential. (ix) Suppose that X is a normed space. Let (e'j)ie/ -> e < oo and ((xi,x*))i€l have the properties from (a) or (b). Taking into account that \X{, 3Jj j

\Xy X / — \X{

Xj X^ J ~\- \X) X* -X*) = (Xi,X* -X*) +

(Xi-X,X*)

for every i, we get ((y — Xi,x*)) —t (y — x,x*) for every y 6 X in both cases. But V i e / , Vy£X

: f(xi) +

(y-xi,x*)
taking the limit inferior, we obtain x* G def(x).

O

In the preceding theorem we obtained that df is a monotone multifunction. In fact df is cyclically monotone. One says that T : X =} X* is cyclically monotone if for all n S N and ((xi,x*))._. C g r T one has Y"

(xi+1-Xi,x*)<0,

(2.36)

*—'i=0

where xn+i := XQ. Taking n = 1 in Eq. (2.36) we obtain that {x\ - x$, x0') + (x0 — xi,x\) < 0, i.e. (xi —xQ,xl — XQ) > 0, for all (X0,XQ), (XI,XI) £ grT. Hence every cyclically monotone multifunction is monotone. When / : X -> R is a proper function, df is a cyclically monotone multifunction. Indeed, let n e N and ((xi,a;*))" =0 C gr<9/; then (xi)™=0 C d o m / and (xi+i -xt,x*)

< f(xi+i)

- f(xi) Vi, 0 < i < n,

where, as above, xn+\ := XQ. Adding these relations side by side for i = 0 , 1 , . . . , n, we get Eq. (2.36). Note that to a nonempty cyclically monotone multifunction one can associate a function / 6 T(X).

The subdifferential

of a convex

85

function

Proposition 2.4.3 Let T : X =4 X* be a cyclically monotone function and (icc^o) £ g r ^ - Consider fr : X —>• M defined by h{x)

:= sup f (a; - z n , < ) + X ) .=Q (a:*+1 ~ ^ ^ ) ) >

where the supremum is taken for all families ((x,,x* ))" = 0 C g r T N. T/ien / T e T(X), /T(XO) = 0 and T(x) C dfT(x) for every x

multi-

( 2 - 37 ) withne eX.

Proof. Because fr is a supremum of continuous afHne functions, fa is lsc, convex and nowhere - o o . Note that domT C d o m / y . Indeed, let (x,x*) G grT. Taking k G N, ((a:i,xj))* =0 C grT, n := fc + 1 and ( x „ , < ) := ( i , ? ) , from Eq. (2.36) we obtain that

E

it-i

_

. {Xi+i 1=0

- Xi, X*) + (X-

Xk,X*k)

+ (X0 - X, X*) < 0,

whence fr{x) < (x — x0,x*). Hence x G d o m / y . In particular the preceding inequality shows that / T ( ^ O ) < 0. Taking n = 1 and (xi,x*) = (XO,XQ) in the definition of fr(xo), we get fr(xo) > 0. Hence / T ( ^ O ) = 0. Therefore

h € T(X). Let now (x,x*) G g r T and x G X . Consider an arbitrary it < fr{x) + (x — x,x*). Then there exist k G N and ((XJ,X*)J C g r T such that _

r—\fc-i

/ x - ( x - x , x * ) < {x-xk,x*k) + 2 ^ . = 0 fci+i -a;»,a;*>Taking n = k + 1 and (x n ,a;*) := (x,x*), the preceding inequality yields n—1

Ei=o (

Xi+1

~XuX^

- -^W-

Letting (i —¥ fr (x) + (x — x, x*), we obtain that fr (x) + (x — x, x*) < fa (x), and so x* G 9/r(S). D In the next theorem we use the convexity of the function / . Theorem 2.4.4

Let f G A(X), x G d o m / and e Gffi+. T/»en:

(i) def(x) = df'e(x, -)(0); moreover, ifx£ differentiate at x then df(x) = {V/(x)}.

( d o m / ) 1 and / is Gateaux

(ii) 7/ / is strictly convex then df is strictly monotone. (hi) / is lsc atx if and only ifd£f(x) ^ 0 for every e > 0; d£f*(x*) ^ 0 if f (x*) eRande>0. (iv) Suppose that f is lsc at x. Then x* G dEf(x) <£> x G def*(x*).

Convex analysis in locally convex spaces

86

Proof. (i) Ii x* £ dsf(x), replacing x by x + tu, with t > 0, in Eq. (2.35), then dividing by t we obtain V«€l

: (u, x ) < inf —

^-^

—

fe{x,u),

and so a;* £ df^(x, -)(0). The converse inclusion follows from the inequality f'e(x, u) < f(x + u) — f(x) + e, valid for every u £ X. If / is Gateaux differentiable at x then f'+{x, •) — V/(aT); the conclusion is obvious. (ii) Let x,y £ d o m / , x / y, and x* € df(x), y* £ df(y). From (i) we have that (y-x,x*)

< f+(x,y-x)

<

f(y)-f(x),

& - y, y*) < f+(y, x-y)<

f(x) -

f(y),

because / is strictly convex. Adding the above two relations we obtain that (y — x, x* — y*) < 0. This shows that df is strictly monotone. (iii) Suppose that / is lsc at x and let us take e > 0; using Theorem 2.3.4 we have f(x) = r(x)

= sup{(z,z*) - f*(x*)

I x* G X*} >

f(x)-e.

Therefore there exists a;* £ X* such that {x, x*)—f*(x*) > f(x)—s, whence x* £ def(x). Conversely, suppose that dEf(x) ^ % for every e > 0. Then V e > 0 , 3x*£X*

: f(x) - e < {x,x*} - f*(x*)

<

f(x),

whence f(x) < f**(x). Since / * * < / < / , it follows that J{x) = f(x), i.e. f is lsc at x. Let x* £ X* be such that f*(x*) £ R and e > 0. By the definition of /*, there exists x £ X such that f*{x*) < (x,x*) — f(x) + e, whence (x,x*-x*)

< (x,x*) - f(x) - f*(x*)+e<

i.e.x£def*(x*). (iv) Since / is lsc at x, we have that f**(x) immediate using Theorem 2.4.2 (ii).

f(x*)

-

f*(x*)+e,

= f{x); the conclusion is •

Remark 2.4-1 Note that for / 6 A(X), x £ d o m / and U £ Mx{x) we have that df(x) = d(f + iu)(x), but def{x) and d£(f + iu)(x) may be

The subdifferential

of a convex

87

function

different for e > 0, i.e. df is a local notion while dsf global one for convex functions.

(with e > 0) is a

Indeed, since f'(x, u) — (f + t[/)' (x, u) for every u £ X, the first remark follows from assertion (i) of the preceding theorem. For the second remark let us consider / € A(R) given by f(x) = x2 and U = [—1,1]. Then ds{f + iu)(0) = def{0) = [-2y/i,2y/e\ for e € [0,1] and de{f + t a )(0) = [-1 -£,l + e]£ [-2y/i,2^/i\ = 0 e /(O) for e > 1. Generally, the formula d(\f)(x) = Xdf(x) is not true for A = 0; this formula, however, is true if x £ (dom/) 1 . In assertions (vii) and (viii) of Theorem 2.4.2 we have equality only under supplementary conditions called generally "constraint qualification conditions;" in Section 2.8 we shall give such conditions. Let A C X be a nonempty convex set and a e A; then diA{a) = {x* eX*\Vx£A

:

(x-a,x*)<0}

= {x* | V x G A : (x,x*) < (a,x*)} = {x* | Vx 6 cone(A - a) :

(x,x*)<0}

+

= —(cone(A — a ) ) = —cone(A — a)+. The set <9M(O) is denoted by N(A;a) and is called the normal cone of A at a S A, while the set cone (A — a) is denoted by C(A;a) (even if A is not convex); evidently, the set C(A;a) is a closed (convex if A is so) cone. Taking into account the relation established above and the bipolar theorem (Theorem 1.1.9), we have that N{A;a) = -(C(A;a))+

and C(A;a) = -(N(A;a))

+

.

It is obvious that N(A; a) = {0} if a £ A*. We observe that x* € N(A; a) \ {0} if and only if Hx.^a^x*) is a supporting hyperplane to A at a and Ac//-,, ,,. The set df(x) may be empty even if / is lsc at x. For example, the function / : R -> E, f(x) := -Vl - x2 for \x\ < 1, f(x) := oo for |a;| > 1 (already considered in Section 2.1) is lsc, finite at 1, but df(l) = 0. Moreover 9 ( 0 - / ) (1) = R_. In the preceding section we have met some situations in which it was easy to compute the conjugate of a function. We shall point such situations for the e-subdifferential, too.

88

Convex analysis in locally convex spaces

Corollary 2.4.5 Let fi : Xi —> R for i E \,n be proper functions and x = (xi,... ,xn) E Yli=i dom/j. Let us consider the function

n

n

.

—

Xi-tR,

•


^n

:= > .

fi(xi),

ande E K+• Then de
9

^ ^ )

£i

- °>

£l H

+ e« = e j •

/n particular

n

n

. dfi(xi). Proof. We have already seen in the preceding section (for n = 2, but the extension is immediate) that y*{x\,... ,xn) = Y12=ifi(xi)- Therefore, by Theorem 2.4.2 (ii), x* E de
O 3 e\,...,

1- ( z „ , < ) + e

en > 0, ei + . . . + en = e, Vi G T~n : /i(xi) + /*(z*) - (xi,x*) < et,

O 3 £i, . . . , 6n > 0, £! + . . . + £„ = £,

Vi G l , n : x* €

dEifi(xi),

whence the conclusion.

•

Corollary 2.4.6 Let f : X -» R be a proper function and A E &(X, Y). If x E d o m / and y E Y are such that y = Ax and (Af)(y) = f(x), then for every e E R+ we have de(Af)(y)=A'-1(def(x)). Proof.

Using Theorem 2.4.2 (ii) we have that

y* E d£(Af)(y)

o (Af)(y) + (A/)*(j,*) < (y,y*) +e & f(x) + f*(A*y*) *> A*y* G def(x)

< (Ax,y*) + e = (x,A*y*) + e &y*E

A*'1

(def(x)).

We used the relation (Af)* = f* o A*, proved in Theorem 2.3.1 (ix).

•

The subdifferential

of a convex

89

function

Corollary 2.4.7 Let / i , . . . , / „ : X -> R (n € N) be proper functions. Suppose that for every i € l , n there exists x~i £ Aora.fi suc/t i/iaf (/lD • • • • / „ ) ( ! ! + . . . + ! „ ) = / i ( l i ) + • " • + /„(x„).

(2.38)

Then for every e € E+ a e ( / i D . . . D / „ ) ( i i + ••• + ! „ ) = [ J { ^ e i / l ( ^ l ) n - - - n 9 e „ / „ ( l „ ) | £ l , . . . , e „ > 0, £X + . . . + £ „ = £ } .

In particular

0(/iD... n/n)(xi + • • • + xn) = a/i(ii) n • • • n 5/„(in). Conversely, if dfi(x~i) Pi • • • (~1 dfn(xn)

^ 0, i/ien relation (2.38) is vafo'd.

Proof. We apply Corollary 2.4.5 to the functions fi,- • • ,fn and Corollary 2.4.6 to / : X n ->• E defined by / ( x i , . . . ,£„) := / i ( x i ) + • • • + fn(xn) and to the operator A : X " —> X defined by A(xi,..., xn) := X\ + • • • + xn. If x* £ 5/i(«i) (~1 • • • n dfn{xn), then Vi £ l , n , V i j G-X" :

(xi-Xi,x*)
hence / I ( X J ) H h fn{xn) < / i ( x i ) ^ 1- fn{xn) for all x i , . . . , x n e X such that xi + • • • + xn = x~i + • • • + xn, and so Eq. (2.38) holds. • The next result shows that the convolution has a regularizing effect. Corollary 2.4.8 Let / i , / 2 e A(X), xt e dom/i for i e {1,2} and x — xi +X2- Assume that (fiDf2)(x) = /i(x"i) + f2(x2) and fiDf2 is subdifferentiable at x. If f\ is Gateaux differentiable at x\ then f\Of2 is Gateaux differentiable atx and V(fiDf2){x) = V / ^ x j ) . Moreover, if X is a normed vector space and / i is Frechet differentiable at xi then /1D/2 is Frechet differentiable at x. Proof. By the preceding corollary we have that d(f\Of2) (x) = df\ (x~i) n 0/ 2 (z 2 )- Let x* £ d(f1nf2)(x). Then (u,x*) < (haf2)(x < fl(xi

+ u) - (AD/aXaf)

+u) + / 2 ( x 2 ) - fi{xi)


- /2(x2)

(2-39)

for every u € X. If fi is Gateaux differentiable at x\ then, from the preceding corollary and Theorem 2.4.4 (i), we have that x* = V / i ( x i ) .

Convex analysis in locally convex spaces

90

Using (2.39) we obtain that (fiOf2)'(x,u) = (w, V/i(a; 1 )) for every u € X, which shows that /iD/2 is Gateaux differentiable at x and V(/iD/2)(x) = V/!(li).

Assume now that X is a normed space and f\ is Frechet differentiable at x\. From the preceding situation we have that /1IH/2 is Gateaux differentiable at x and V(/iD/ 2 )(5;) — V/i(afi). Using again Eq. (2.39), we have that 0 < (/id/ 2 )(af + u) - (fiDf2)(x)

- (u,V/i(ii)>

< fi{xi + u) - / i ( i i ) - (u, V/i(ii)>

Vu G X;

hence /1CH/2 is Frechet differentiable at 2; and V(/iD/2)(af) = V/i(aFi). D In Section 2.6 we shall extend the results of Corollaries 2.4.6 and 2.4.7 to the case where the infimum is not attained in y and x, respectively. The following result establishes a sufficient condition for the subdifferentiability of a convex function; this result is of exceptional importance. Theorem 2.4.9 Let f G A(X). If f is continuous at a; G d o m / , then def(x) is nonempty and w* -compact for each e G M+. Furthermore, for every e > 0, f'e(x, •) is continuous and V«6l

: f'e{x,u)

= max{(«,x*) | x* G d£f(x)}.

(2.40)

Proof. Suppose that / is continuous at x G d o m / (from Theorem 2.4.4 we already know that def(x) ^ 0 for e > 0!). Let rj > 0. Since / is continuous at x, there exists V G K x such that Vxex

+ V : f{x)
(2.41)

Therefore (aT + V) x [f(x) + n, oo[C e p i / , whence int(epi/) ^ 0. The set e p i / being convex and (x, f(x)) £ int(epi/) (because (x,f(x) — 5) $. e p i / for every S > 0), we can apply a separation theorem; so, there exists (a;*,a) G X* x E \ {(0,0)} such that V ( z , f ) € e p i / : (x,x*) + at < (x,x*) + af(x).

(2.42)

Taking x — x and t = /(aT) + n, n G N, we get a < 0. If a = 0, using Eq. (2.42), we obtain that {x — x,x*) < 0 for every x G d o m / , and so, by Eq. (2.41), we have that (a;,a;*) < 0 for every x GV; since V is a neighborhood of the origin, this implies that x* = 0. Thus we obtain the contradiction:

The subdifferential

of a convex

function

91

(a;*,a) = (0,0). Therefore a < 0; so we can consider a = - 1 in Eq. (2.42). This relation becomes V x e d o m / : (x,x*) - f(x) < (x,x*) — f(x), i.e. x* e df(x). Let now e > 0. For every x* G dsf(x) we have V i i g y : (u,x*)

= df£(x, -)(0), using Eq. (2.41),

+ u)-f(x)+£
+ e,

(2.43)

whence {u,x*) > - 1 for every u £ (r] + e)~1V (recall that V is symmetric) which proves that def(x) C (fa + e ) - 1 V)° = (r, + e)Va.

(2.44)

By the Alaoglu-Bourbaki theorem (Theorem 1.1.10) we have that V° is w*compact; since def(x) is w*-closed, it follows that d£f(x) is w*-compact. Taking into account that x £ int(dom/), from Theorem 2.1.14, we have that domf^x, •) = X; hence, using Theorem 2.2.13 and Eq. (2.43), the function fg(x,-) is continuous. Furthermore, using again Eq. (2.43), we have that f'e(x,u)

> sup{(u,a;*) | x* £

def(x)}.

We intend to prove that in the above inequality we have equality and the supremum is attained. For this end let u € X \ {0}. Consider XQ := Eu and ip : Xo -> E, p(tu) := tf'e(x,u); it is obvious that ip(u') < f'E(x,u') for every u' 6 Xo, and so, applying the Hahn-Banach theorem, there exists x* 6 X' such that (u,x*) = tp(u) = f^{x,u) and (u',x*) < f'e(x;u') for every u' € X. Since f^(x,-) is continuous, x* is continuous, too; hence x" e df'£{x, -)(0) = dEf(x). The proof is complete. • Using the preceding result we obtain the following criterion for the Gateaux differentiability of a continuous convex function. Corollary 2.4.10 Let f € A(X) be continuous atxe domf. Then f is Gateaux differentiate at x~ if and only if df(x) is a singleton. Proof. The necessity was already observed in Theorem 2.4.4(i). Assume that df(x) = {x*}. Using Theorem 2.4.9 we obtain that f'(x,u) = (u,x*) for every u € X. Because lim^o* - 1 (f(x + tu) — f{x)) = —f'(x,—u) =

Convex analysis in locally convex spaces

92

— (—u,x*) = (u,x*), we have that / is Gateaux differentiable and Vf(x) = x*. D Related to Frechet differentiability of convex functions see Theorem 3.3.2 in the next chapter. When / is not continuous at x £ d o m / , relation (2.40) may be false; see Corollary 2.4.15 and Exercise 2.30. However the following result holds. Theorem 2.4.11

Let f £ T(X), x £ dom / , and e £ ]0, oo[. Then

Vu£X

: f'e(x,u)=sup{{u,x*)\x*

£d£f(x)}.

(2.45)

Therefore f'e(x, •) is a Isc sublinear function. Furthermore, for every u £ X, f'(x,u)

= lim (swp{{u,x*) | x* £

def(x)})

= inf (sup{(u,a:*) | x* £ def(x)}). Proof. whence

(2.46)

In Theorem 2.4.4(i) we have seen that dEf(x)

V / 6 3E/(f),Vii6l

: {u,x*) <

—

df^.(x,-)(0),

ft(x,u).

Therefore the inequality " > " holds in Eq. (2.45). For proving the converse inequality, let u £ X and A £ R with A < f'£(x,u). From the definition of f'e (x, u), we have that Vt>0:X
+ t u )

;

m + £

(2.47)

and A

<

lim

f^

t—>oo

+ tu)-f(x)+e t

=

^ (—>oo

f(x

+

tu)-m

= /oo(u),

(2.48)

t

Let us consider A := e p i / and B := {(x + su, f(x) + Xs — e) \ s > 0}. It is obvious that A and B are nonempty closed convex sets and, from Eq. (2.47), A n B = 0, i.e. (0,0) ^ A — B. Moreover B is locally compact (as subset of a finite dimensional separated locally convex space). Since Aoo = epi /oo and B00=R+(u, A), from Eq. (2.48) we obtain that 4oonBco = {(0,0)}. Then, by Corollary 1.1.8, A — B is a closed set. Applying Theorem 1.1.7, there exists (x*,a) 6 X* x E such that V(x,t) £ epi/, V s > 0 : 0 > (x - x-

su,x*) + a(t - f(x) - sX + e).

The sub differential of a convex function

93

Taking x = x and letting t -> oo we obtain that a < 0; if a = 0, for s = 0 we get the contradiction 0 > 0. Therefore a < 0 and we can suppose that a = — 1. The above relation becomes V x G d o m / , V s > 0 : 0 > (a; -x,x*)

- (/(«) - /(a?) + e ) +s(A - ( u , i * » .

For s = 0 we obtain that x* G dsf(x), while for s -> oo we obtain (u,a7*) > A. Therefore A < sup{(u,a;*) | a;* £ 5 e /(aT)}. Since A < f^(x,u) is arbitrary, the relation (2.45) is true. The equality (2.46) follows from relation (2.45) and Theorem 2.1.14. • The subdifferentiability criterion of Theorem 2.4.9 can be extended. Theorem 2.4.12 Let f G A(X) and X0 := aff(dom/). / / f\Xo is continuous atxG domf, then df(x) ^ 0. In particular, if dim X < oo, then df(x) ^ 0 for every x G *(dom/). Proof. Without any loss of generality, we can suppose that x = 0; then X0 = lin(dom/). The function g := f\x0 is convex, proper and continuous at 0. By Theorem 2.4.9 we have that dg{0) # 0. Let v? G dg{0); hence ip € XQ and V x g d o m j : tp[x) — (p(0) < g(x) — g(0). Using the Hahn-Banach theorem we get i* G X* such that x*\x0 = tp. The above inequality shows that Vx G d o m / = dome? : {x - 0,x*) = tp{x) < g(x) - g(0) = f(x) - / ( 0 ) , whence x* G 9/(0). If dimX < oo, then dimXo < oo. By Theorem 2.2.21, g is continuous on int(dom/) = l ( d o m / ) . The conclusion follows from the first part. • Note that under the conditions of Theorem 2.4.12 df(x) is, generally, not bounded, and so it is not «;*-compact. But, under the conditions of Theorem 2.4.9, in the case of normed spaces, df has supplementary properties (mentioned in the next theorem). The local boundedness of monotone operators was proved by R.T. Rockafellar (see Theorem 3.11.14); the converse is true in Banach spaces for lower semicontinuous functions as shown by Corollary 3.11.16. Theorem 2.4.13 Let X be a normed space, f G A(X) be continuous on int(dom/) and e > 0. Then dsf is locally bounded on int(domc?/) =

94

Convex analysis in locally convex spaces

int(dom/). Moreover, if f is bounded on bounded sets then f is Lipschitz on bounded sets and dsf is bounded on bounded sets. Proof. By Theorem 2.4.9 we have that int(dom/) C dome?/ C d o m / ; hence int(dom<9/) = int(dom/). Let xo £ int(dom/); since / is continuous at x0, from Corollary 2.2.12 we get the existence of m, p > 0 such that Vx,y€D(x0,2p)

: \f(x) - f(y)\ < m\\x - y\\.

Let us fix a; € D(xo,p). Then for y £ x + pUx we have that f(y) < f(x) + mp, i.e. Eq. (2.41) holds with V := pUx and r] :— mp. Using Eq. (2.44), we obtain that def(x) C (r? + e)V° = (m + e/p)Ux* for every x £ D(x0,p). Assume now that / is bounded on bounded sets. Then d o m / = X. Taking p > 0, / is bounded above on 2pUx, and so Eq. (2.41) holds with V := 2pUx and some r\ > 0. By Corollary 2.2.12 / is Lipschitz on pUx, and, by Eq. (2.44), def(x) C (?? + e)p~1Ux* for x e D{x0,p). The proof is complete. • For other relationships between the continuity of / and the local boundedness of df see Corollary 3.11.16. In Theorem 2.4.4 we have seen that finding the e-subdifferential of a convex function at a point reduces to compute the subdifferential of a sublinear function at the origin. Other subdifferentials (the subdifferentials of Clarke, of Michel-Penot, etc.), for nonconvex functions are introduced through certain sublinear functions. So, we consider it is worth giving some properties of sublinear functions. T h e o r e m 2.4.14 Let f,g:X->M.,h:Y->Rbe sublinear functions, T £ &(X, Y) and B,C C X* be nonempty sets. Then: (i)

/ * = ^9/(0);

(ii) 5/(0) ^ 0 o / is Isc at 0; (hi) for every x G dom / and for every e £ R+ we have:

dm

= {x*edf(p)\(x,x')

= f(x)},

d£f{x) = {x* e 3/(0) I (x,x°) > f(x) - £},

def(0) = 0/(0);

(iv) if f is Isc at 0 then Vxel

: J(x)=sup{{x,x*)\x*

edf{0)}.

The subdifferential

of a convex

function

95

(v) Suppose that f and g are Isc. Then f < g <3> 9/(0) C dg(0). (vi)

The support function SB of B is sublinear, Isc, SB = (<-B)* and — coB, the closure being taken with respect to the weak* topology w* on X*; moreover, SB < sc if and only if B C coC. 0SB(O)

(vii) / / h is Isc then d(h o A)(0) = w*-c\ (A*{dh{0))); (viii)

if f and g are Isc, then d(f + g)(Q) = w*-c\ (3/(0) + dg(0)).

Proof,

(i) For every x* € X* we have

f*(x*)

= s u p ^ O - f{x) I x € d o m / } > (0,a;*) - /(0) = 0.

If a;* £ df(0) then (x,x*) < f(x) for every x £ d o m / , whence f*(x*) = 0. If x* £ <9/(0), there exists x e X such that (x, x*) > f(x). In this situation we have f*{x*) > sup{(tx,x*)

- /(tx) | t > 0} = sup{i(x,a;*) - /(x) | t > 0} = oo.

(ii) If 9/(0) ^ 0 then, from Theorem 2.4.1(h), we have that / is Isc at 0. Suppose that / is Isc at 0. Then /(0) = /(0) = 0, and so, by Theorem 2.3.4, /(0) = /**(0) = 0. Assuming that <9/(0) = 0 we obtain that /* = £9/(0) = +00, and so the contradiction /** = — 00. Therefore 9/(0) ? 0. (iii) Let x £ d o m / and e > 0. If x* £ 9/(0) and (a;,a;*) > /(a;) — e then WxeX

: {x-x,x*)

= {x,x*)-(x,x*}


i.e. x* £ d£f(x). Conversely, let x* £ def(x); then the above inequality holds. Taking x = 0 we get (af, a;*) > f(x) — e; taking now x :=x + ty, t > 0, y £ X, we obtain that *(l/,**) < / ( i + ty) - f{x) +e<

fix) + tfiy) - fix) + e = tf(y) + s.

Dividing by t > 0 and letting then t -> 00, we obtain that (y,x*) > fiy) for every y € X, i.e. x* £ 9/(0). For e = 0 we have 0/(3:) = {x* € 0/(0) I (a;,a:*) > fix)} = {x* £ 0/(0) | (x,x*) = while for x = 0 we obtain that 9 £ /(0) = K

G 9/(0) I (0,ar*) > /(0) - e } - 9/(0).

fix)},

96

Convex analysis in locally convex spaces

(iv) Suppose that / is lsc at 0. Then /(0) = /(0) = 0. Using Theorem 2.3.4, we have that /** = J. Therefore J(x) = sup{(a;,a;*) - f*(x*)

| x* 6 X*} = sup{(x,x*)

\ x* € 0/(0)}.

(v) It is obvious that 0/(0) C dg(0) if / < g. The converse implication follows immediately from (iv). (vi) At the end of Section 2.3 we noted that SB is lsc, sublinear and sB = {LB)*'• From the obvious inclusion B C 8SB(0) we get coB c <9SB(0) (because <9SB(0) is convex and w*-closed). Let x* fi coB. Using Theorem 1.1.5 in the space (X*,w*) and taking into account that (X*,w*)* — X, there exist x E X and A s K such that Vx* G coB D B : (x,x*) > A >

(x,x*),

whence (x,x*) > ssi'x), i-e. x* £ 8SB(0). The conclusion follows. (vii) By Theorem 2.4.2 we have that A*(dh(0) C d(h o A)(0). Let x* i w*-c\ (A*(dh(0))). Using Theorem 1.1.5, there exist x € Y and A e R such that (x,x*) > A > (x,Ay*) — {Ax,y*) for every y* e dh(Q). From (iv) we get A > h(Ax), and so x* $. d(h o A)(0). (viii) Let H : X x X -> 1 , H(x,y) := f(x)+g(y), andT:I->XxX, Tx := (x,x). It is obvious that H is sublinear and lsc, and T is continuous and linear; moreover, dH(0,0) = 5/(0) x dg{0) and T*(x*,y*) = x* + y*. By (vii) we obtain that d(f + g)(0) = d(H o T)(0) = «>*-cl (T*(dH(0,0)))

= w*- cl(5/(0) + dg(0)).

The proof is complete.

•

Using the preceding result we have Corollary 2.4.15 Let f G \(X) and x G d o m / . Then df(x) ^ 0 if and only if f'(x, •) is lsc at 0. In this case f'{x,-)(u)

= sup{(u,x*) | Z* G df(x)}

Vuel.

Proof. The conclusion follows from the formula df(x) — df'(x, -)(0) (see Theorem 2.4.4) and assertions (i) and (iv) of the preceding theorem. • Note that, even for X = E 2 and / G T(X) subdifferentiable at x, f'(x,-) may not be lsc; take for example / := i\j and x = (0,1), where U:={(x,y)em2 \x2+y2
The sub differential of a convex

function

97

Corollary 2.4.16 Let (X, || • ||) be a normed space, f : X -> R, f(x) := \\x\\, andx G X. Then, denoting Ux* by U*, we have: r Proof.

= iu;

df(0) = ir,

df(x) = {x*eU*\(x,x*)

= \\x\\}.

The above formulas follow immediately from Theorem 2.4.14. •

Another example of sublinear function is given in the following result. Corollary 2.4.17 Let / : E n ->• E, f(x) := xi V • • • V xn, and x e W1, where n G N. Then 5/(0) = A n , / * = IA„, def(x)

= {y e An\xiyi

+ ...+xnyn>

f(x)-e},

where A » : = { ( A i , . . . , A n ) 6 " B I A i > 0 , Ai + . . . + A„ = 1}. Proof.

It is obvious that / is sublinear and continuous. Moreover y G <9/(0) » V x G C

: xlVl

+ ••• + xnyn

< n V • • • V xn.

Taking Xj = 0 for j ^ i and Xi = —1, we obtain that — j/» < 0, i.e. yi>0 for every i. Taking then X{ — t G K for every i, we obtain that £(j/H \-yn) < t for every i, whence y±-\ \-yn = l. Therefore 9/(0) C A n . The converse inclusion is immediate. The other relations follow directly from Theorem 2.4.14. • The following theorem furnishes a formula for the subdifferential of a supremum of convex functions. Other formulas will be given in Section 2.8. Theorem 2.4.18 (loffe-Tikhomirov) Let (A,T) be a separated compact topological space and fa : X —> E be a convex function for every a G A. Consider the function f := supaGAfa andF(x) := {a G A | fa{x) = f(x)}. Assume that the mapping A 3 a 4 / „ ( i ) G E is upper semicontinuous and XQ G dom / is such that fa is continuous at XQ for every a £ A. Then df(x0)=co(\J

dfa(x0)).

(2.49)

Proof. Since A is compact and a H-> fa{x) is upper semicontinuous we have that F{x) is a nonempty compact subset of A for every x G X. The inclusion D being true without any condition on A and / (and easy to prove), let us show the converse inclusion. If f(xo) — —oo, then fa(xo) =

Convex analysis in locally convex spaces

98

—oo for all a £ A = F(x0), and so df(x0) = dfa(x0) — 0; the conclusion holds. We assume now that f(x0) £ R. Let us note first that in our conditions x0 £ (dom / ) ' . Indeed, let x £ X. Consider a fixed 70 > f(xo) (> fa(xo))Because fa is continuous at xo, for every a £ A there exists ta > 0 such that fa(xo + tax) < 70. Since /? t-> /^(^o + tax) is upper semicontinuous, the set Aa := {/3 £ A \ fp(x0 + £Q:r) < 70} is an open set containing a. As A = UaeA-^<*> there exist a i , . . . , an £ A such that A = \J™=1 Aai. Let t := m i n { £ a i , . . . , i Qn } > 0. It follows that fa(x0 + tx) < 70 for every a £ A, and so f(xo + tx) < 70. Hence x0 £ ( d o m / ) \ Since fa is continuous at xo, dfa(xo) 7^ 0 for every a £ F(xo), and so Q ^ 0, where Q is the set on the right-hand side of Eq. (2.49). Suppose that there exists x* £ df(x0) \ Q- Using Theorem 1.1.5 in the space (X*,w*), there exist x £ X and e > 0 such that VaeF(io),Vi'e9/a(i0)

: (x,x*) - e > (x,x*),

(2.50)

or equivalently, by Theorem 2.4.9, V a e F ( i „ ) : (x,x*)-e>{fa)'{x0;x).

(2.51)

Because x0 £ (dom/) 1 , we may suppose that x0+x £ d o m / . Let t £]0,1[; then xo + tx £ dom / . There exists at £ A such that fat (XQ + tx) — f(x0 + tx). Because (1 - t)fa,(x0) + tfat(x0 + x) > fat(x0 + tx) and x* £ df(x0), we obtain that (1 - t)fat (Xo) > f(x0

+ tx) - tfat (x0 + X) > f(x0

+ tx) - tf(x0

+ x)

> f(xo) - t (x,x*) - tf(x0 + x). It follows that limt^o fat(xo) = f(xo). The space (A,T) being compact, there exists a convergent subnet (a^^j^j of (a t ) t e ] 0 ,i[ converging to ao £ A; hence fao(x0) = f(x0), i.e. a0 £ F(x0)- Let s e]0,1[ be fixed (for the moment) and take t £ ]0, s]. Using Eq. (2.50) we obtain that fat(xo

+SX) - fat(x0)

s

fqt(x0

>

+ tx) - fgt{x0)

t

f(x0

-

+ tx) - f (x0)

t

(x,x*).

We have that tp(j) < s for j y j s for some j s £ J. Taking t = ip(j) with

The general problem of convex

programming

99

j y j s and using the upper semi-continuity hypothesis, we obtain that Va€]0>i[:

Ui^ +

^-Ui^) s

and so (jao)'{xo;x) > (x,x*), contradicting Eq. (2.51). D Of course, for conjugates and for the e-subdifferentials it is desirable to dispose of numerous formulas. There exist effectively a large set of such formulas we shall establish in Section 2.8. 2.5

The General Problem of Convex Programming

By problem of convex programming we mean the problem of minimizing a convex function / : X —> R, called objective function (or cost function) on a convex set C C X called the set of admissible solutions, or set of constraints. We shall denote such a problem by (P)

min f(x),

xeC.

Of course, to consider this problem, X must be a linear space, however most of the results will be obtained in the framework of separated locally convex spaces or even normed spaces. To problem (P) we can associate a problem (apparently) without constraints: (P)

min f(x),

x G X,

where / := / + vein order for the problem (P) to be nontrivial, it is natural to assume that C n dom / ^ 0 (<£>• dom / ^ 0) and that / does not take the value — oo on C (i.e. f does not take the value — oo). We call value of problem (P) the extended real v(P) := v(f,C)

:= m£{f(x)

| x G C} £ I ;

we call (optimal) solution of problem (P) an element x G C with the property that f(x) = v(P); this means that x is a (global) minimum point for the function / . We denote by S(P) or S(f, C) the set of optimal solutions of problem (P). Therefore S(P) = {x G C | Vx G C : f(x) < f(x)} = {zGX|VxGX

: f(x) < f(x)} = S(P)

Convex analysis in locally convex spaces

100

if C fl dom / ^ 0. The set S(f, X) is denoted also by argmin / . Of course, an important problem is that of the existence of solutions for (P), resp. (P). The most important result which assures the existence of solutions for (P) is the famous Weierstrass' theorem. Because the underlying spaces are not compact we have to use some coercivity conditions. We say that / : X —> E is coercive if limn^n^oo f(x) — oo. It is obvious that / is coercive if and only if all the level sets [/ < A] are bounded (see also Exercise 1.15); when / is convex then / is coercive if and only if the level set [/ < A] is bounded for some A > inf / (see Exercise 2.41). We have the following result. Theorem 2.5.1 (i)

Let f £ T(X).

If there exists A > v(f,X)

such that [f < A] is w-compact,

then

s{f,x)^d>. (ii) If X is a reflexive Banach space and f is coercive then S(f,X)

^ ill.

Proof, (i) Of course, v(f,X) = v(f, [f < A]). Since / is lsc and convex, / is tu-lsc. The conclusion follows using the Weierstrass theorem applied to the function / | [ / < A ] -

(ii) Because / is coercive (see Exercise 1.15), [/ < A] is bounded for every A e E. Since [/ < A] is w-closed and X is reflexive, we have that [/ < -M i s w-compact for every A G E. The conclusion follows from (i). • Of course, in the preceding theorem, the condition that / is convex can be replaced by the fact that / is quasi-convex. The next result shows that the reflexivity of the space X is almost necessary in Theorem 2.5.1. Theorem 2.5.2 Let (X, ||-||) be a Banach space. Assume that there exists a proper function f : X —> M. satisfying the following conditions: (i) (ii)

[/ < f(x)] l'5 closed, convex and bounded for every x € dom / ; / attains its infimum on every nonempty closed convex subset of

X; (iii) there exist xo,xi £ d o m / and r > 0 such that f is Lipschitz on [f < /On)] and [f < f(x0)] + rUx C [/ < / f a ) ] , where Ux := {x € X |

IMI < 1}Then X is reflexive. Proof. Of course, f(x0) < f(xi); if f(x0) = f{xi) then the relation [/ < f(xo)] + rUx C [/ < f(xi)] contradicts the boundedness of [/ <

The general problem of convex programming

101

f(xi)]. Hence f(x0) < f(xi). Since / is Lipschitz on [/ < f(x\)], there exists L > 0 such that \f(x) - f(x')\ < L\\x-x'\\ for all x,x' £ [f < f(x\)]. Since /|[/ 0 such that allx C A c /SC/x- It follows that a - 1 1 | - | | = paux > PA > Ppux = P~x ||-||, where p ^ is the Minkowski gauge associated to A. By Theorem 1.1.1 PA is a semi-norm, and so it is a norm equivalent to ||-||; moreover, by Proposition 1.1.1, the unit closed ball with respect to PA is cl A To obtain that X is reflexive, by the famous James' theorem (see [Diestel (1975), Th. 1.6]), it is sufficient to show that every x* € X* attains its infimum on A. Let x* £ X* \ {0} and a* := inf{(x,x*) \ x £ S}. Because S is bounded, a* € E. Let H := {x £ X \ (x,x*) = a*}. It is obvious that (S+eUx)nH # 0 for every e > 0. Taking e 6 ]0, S0] and xe £ (S+eUx)r\H, we obtain that f(xe) < f{x2) + eL, and so mixen f(x) < f(x2). By (ii) there exists xi £ H such that f(x~i) = inf x € ij f(x), and so x\ £ Sf)H. Therefore (xi,x*) < (x,x*) for every x £ S. Similarly, there exists x2 £ S such that (x2,x*) > (x,x*) for every x £ S, whence there exists x := Xi —x2 £ A such that (x,x*) < (x,x*) for every x £ A. • Also the coercivity condition in Theorem 2.5.1(h) is essential as the next theorem will show. In order to establish it we introduce some preliminary notations and results. Let / 6 T(X) and denote by 11/ the set {g £ T(X) | domg = d o m / , supx€domf

\f(x) - g(x)\ < oo, inf / = inf g}.

Since for g £ Hf we have that / — 7 < / < / + 7for some 7 £ M+, we have that /oo = g^ and / , g are simultaneously coercive or not coercive. Consider d : Ilf x Ilf ^ M+, Lemma 2.5.3 metric space.

d{gug2)

:= s u p x 6 d o m / \f{x) - g{x)\ .

The mapping d is a metric on 11/ and (11/, d) is a complete

Proof. It is obvious that d is a metric. Let (gn)n>i C 11/ be a Cauchy sequence. Then (gn{x))n>1 C E is a Cauchy sequence, and so (gn(x)) ->

102

Convex analysis in locally convex spaces

g(x) E R for every x G d o m / . Moreover, ( s u p x e d o m / \gn(x) - g(x)\)n^,1 -> 0. We extend g to X setting g(x) := oo for x G X \ dom / ; hence dom g — dom / . Because (gn(x)) —> g(x) for every a; G X, from Theorem 2.1.3(h) we have that g is convex. Let e > 0; then there exists nE such that <7„(a;) — s < g(x) < 9n(x) + £ for all n > nE and a; G d o m / . It follows that g is proper (being minorized by gnc — e even on X) and inf / — e = inf gn — e < inf 0 which shows that inf g = inf / (even if inf/ = - c o ) . We must only show that g is lsc. Take first x G domg = d o m / and A < g{x). There exists e > 0 such that A < g(x) — 2e. As above, there exists n = ne such that g(y) — e < gn(y) < g(y) + £ for every y G d o m / . In particular g(x) — e < gn(x). Because gn is lsc at x, there exists U G Mx{x) such that 0 andy E X. Then the se£co(epi/U{(y,inf / — e)}) is i/te epigraph of a function fyi£ E T(X) with inf fViS = inf / — e and argmin / y>£ = y + K. Moreover, if f{y) < inf / + e then f — 2e < fy<£ < f, and so g := / y i £ + e G 11/ and d(f,g) < e. Proof. Consider a := inf / — e < inf / . Denote A := co (epi / U {(y, a)}) and let (x, t) E A and s > t. We want to show that t > a and (a;, s) E A. If these happen then epiipA = ^4 [VA being defined in Theorem 2.1.3(iv)], fy,e := VA £ r ( X ) , / y , £ (y) = a = inf /j, ) £ and fy>e < / . Moreover, if f(y) < inf / + e, then f(y) — 2e < a, which shows that (y, a) E epi(/ — 2e). As / — 2e < f, we obtain that A C epi(/ - 2e), and so / - 2e < / y ) £ . Indeed, there exist the nets (Ai)igj C [0,1] and ((xi,ti))i€l C e p i / such that (a;,*) = lim, e / (\i(xi,U) + (1 - Aj)(2/,a)), and so x - lim i 6 / (XiXi + (1 — Aj)y) and t = limjg/ (A;£; + (1 — Aj)a). Since [0,1] is a compact set, we may suppose that (A;) i e / -> A G [0,1]. Of course, A;£; + (1 - Xi)a > Ajinf / + (1 - \i)a > a for every i E I, and so i > a. If (x,i) = {y,a) then limn^oo (^(x 0 , to +ns- na) + n^{y, a)) = (x, s), and so (x, s) E A,

The general problem of convex

programming

103

where (xo,to) is a fixed element of e p i / . In the contrary case Aj > 0 for i >z_ j 0 for some i 0 £ I. Then limi^ io (Xi(xi,ti+X^1(s-t))+ (1-Xi)(y,a)) = (x, s), and so (a;, s) £ A. Let u £ K; then (u,0) £ (epi/)oo- Fixing (xo,io) £ e p i / , we have that (x0 + nu,io) £ e p i / for every n € N, and so ^(zo + nu, to) + ^^{y, a) £ A for every n £ N. Taking the limit we obtain that (y + u, a) £ A, whence fy<s(y + u) < a. Hence y + uG argmin/ y>£ . Conversely, let z € argmin/ y , £ . Assume that u := z - y ^ 0. It follows that (z,a) € A, and so there exist the nets (A;)j € / C [0,1] and ((xi,ti))ieI C e p i / such that (2,a) = lim i 6 / (Aj(x;,£;) + (1 - Xi)(y,a)). Because z ^ y, A» > 0 for i y i0 (for some io € / ) • As above, we may assume that (A;)j 6 / ->• A € [0,1]. If A ^ 0 then ((xj,ii)) i g / ->• (A-1.z + (1 - X~1)y,a), a contradiction because ti > inf/ > a. Hence A = 0. It follows that limjg/ A; {xi,t{) = (u,0). Let (a;,i) £ e p i / . Then Aj (xi,U) + (1 - A,)(a;,i) € e p i / , and so, taking the limit, we obtain that (x, t) + (u, 0) € epi / . This shows that (u, 0) £ rec(epi/) = (epi/)oo- Therefore u £ K. Hence argmin/y ]E = y + K. • Theorem 2.5.5 Let X be a reflexive Banach space and f £ T(X) be such that K n -K - {0}, where K := {u £ X | /<»(«) < 0}. Then the following statements are equivalent: (i) / is not coercive, (ii)

there exists g £Uf

such that argming = 0,

(iii)

{g &Uf \ argming = 0} is a dense G$ set (see page 34)-

Proof. It is obvious that (iii) => (ii). (ii) =>• (i) Let g SUfbe such that argming = 0. From Theorem 2.5.1(h) we have that g is not coercive. Since g £ Uf, there exists 7 > 0 such that 9 — 7 < / < 5 + 7> which implies that / is not coercive, too. (i) =*- (iii) First of all note that for any g £ Uf, g is not coercive and /oo = Soo (because / - 7 < P < / + 7for some 7 € M+). If inf/ = - 0 0 then Q := {g £ Uf \ argming = 0} = Uf, and so the conclusion holds. Let inf / £ R. Without loss of generality we assume that inf / = 0. For every n £ N consider the set Gn •= {g e Uf I min x e n C / x g(x) > inf g = 0}. The use of min instead of inf is possible because g is w-lsc and nllx is w-compact. Also note that Qn is open in (Uf,d); just take g € Qn and

104

Convex analysis in locally convex spaces

0 < 5 < mixenUx g(x); then the ball B(g,8) C Qn. As Q = f]n€^Gn, it is sufficient to show that Qn is dense for any n > 1. For this fix n > 1, e > 0 and /i G 11/. We may assume that /i(0) < e (otherwise replace /i by h(xo + •)> where xo is taken such that h(xo) < e). There exists y G X such that h(y) < e and (y + if) n nUx = 0- Indeed, when K = {0} take y € X \ nUx such that h(y) < e; this is possible because the set [h < e] is not bounded (see Exercise 2.41). Assume now that K ^ {0}. Then there exist z G K and r > 0 such that (z + if) n rUx = 0, or equivalently z ^ r[/x — K. Otherwise K C fl r >o (rUx — K) C —K, a contradiction. The last inclusion is obtained as follows: consider z e Hr>o (rUx - K)\ then z = ^un — kn with un e Ux and kn € if, for every n € N, and so (fc„) ->• —z € if. Consider g := /iy]£ +e. By Lemma 2.5.4 we have that g G 11/, d(/i, g) <e and argmin/i = y + if. Since (2/ + if) D nUx = 0, we have that g G QnTherefore Qn is dense in 11/. • Note that the reflexivity of the space was used in the proof of the preceding theorem only to ensure that the infimum of g on nUx is attained; so the preceding result remain valid when working on the dual of a normed space, the considered convex functions being w*-lsc. Also note that the condition Xo := if fl—if = {0} in the preceding theorem is not essential; if Xo 7^ {0} one obtains a similar result taking into account the constructions in Exercise 2.24. For a similar result when X is not a normed space see Exercise 2.25. Another important problem in optimization theory is the uniqueness of the solution when it exists. The following result gives an answer to this problem. Proposition 2.5.6 Let f G A(X). Then S(f,X) is a convex set. Furthermore, if f is strictly convex then S(f, X) has at most one element. Proof. Let x G S(f,X); then S(f,X) = [f < f(x)], whence S(f,X) is convex. Let now / be strictly convex, and suppose that S(f,X) contains (at least) two distinct elements xi and xi\ since / is proper, S(f,X) C d o m / . Then we obtain the contradiction

v(f,X) < f (\Xl + |x 2 ) < i/On) + \f[x2) = v(f,X). Therefore S(f, X) has at most one element.

•

The general problem of convex

programming

105

As we already know, the practical method for determining extremum points of a function is to determine the points which verify the necessary conditions then to retain those which verify the sufficient conditions. In convex programming we have a very simple necessary and sufficient condition for optimal solutions. Theorem 2.5.7 If f £ A(X), then x E dom / is a minimum point for f if and only if 0 E df(x). Proof. Indeed, f(x) < f(x) for every x E X if and only if x £ dom / and 0 < f(x) — f(x) for every x £ X, which means that 0 € df(x). O Therefore, in convex programming, the minimum necessary condition is also a sufficient condition. In optimization theory local optimal solutions also play an important role; if g : X —>• E, we say that x £ X is a local minimum (resp. local maximum) point if there exists V € Nx{x) such that f(x) < f(x) (resp. f(x) > f(x)) for every x £ V. The convex programming problems present a particularity. Proposition 2.5.8

Let f : X —• E be a convex function.

(i) Ifx(z dom f is a local minimum point for f, then x is also a global minimum point; (ii) if x € dom / is a local maximum point for f, then x is a global minimum point for f. Proof, for every therefore y := (1 -

(i) By hypothesis there exists V £ Nx(x) such that f(x) < f(x) x € V. Suppose that there exists x £ X such that f(x) < f(x); f(x) £ E. Since V £ J^x(x), there exists A £]0,1[ such that X)x + Xx £ V. Therefore /(5?) < f(v) = / ( ( I - A)3F + Ax) < (1 - A)/(3F) + Xf(x),

whence the contradiction f(x) < f(x). Therefore x is a global minimum point for / . (ii) By hypothesis there exists U £ J^cx such that f(x + x) < f(x) for every x £ U. If f(x) = — oo, it is clear that x is a global minimum point of / . Suppose that f(x) £ E (hence / is proper because x € int(dom/)). Then

Vx€U

: f{x) = f{\{x + x) + \{x-xj)

<\f{x + x) +

\f{x-x)
106

Convex analysis in locally convex spaces

Therefore f(x) = f(x) > f(x) for every x € x + U € N j ( x ) . Hence a; is a global minimum point of / . D This result explains why in a convex programming problem we look only for global minimum points. In practical problems, solved numerically on computers, frequently it is not possible to determine the exact optimal solutions (because one works with approximate values). A simple example in this sense is the problem of minimizing the function / : E —» R, f{t) := (t — 7r)2. Taking into account this fact, the notion of approximate solution is proved to be useful. More precisely, if e e R+, we say that x £ C is an e- (optimal) solution of problem (P) if f(x) < f(x) + e for every x 6 C; we denote by SS(P) or Ss(f,C) the set of e-solutions of problem (P). It is obvious that when C fl dom / / 0 we have that Se(f, C) = Se(f, X); moreover, if / is proper and S£{f,C) ^ 0, then v(f,C) G R and Se(f,C) = {x £ C | f(x) < v(f, C) + e}. Related to e-solutions we have the following result. Proposition 2.5.9 Let f £ A(X), x e d o m / and £ € P. Then Se(f,X) is convex; the set Se(f,X) is nonempty if f is bounded from below. Furthermore, x € Sc(f,X) if and only if 0 6 d£f(x). •

2.6

Perturbed Problems

We shall see (especially) in the following two sections that it is very useful to embed a minimization problem (P)

min f(x),

x 6 X,

in a family of minimization problems. In this section X, Y are separated locally convex spaces if not stated explicitly otherwise and / : X —> R. Let us consider a function $ : I x F 4 l having the property that f(x) — $>(x, 0) for every x € X; $ is called a perturbation function. For every y £ Y consider the problem (Py)

min $(x,y),

x 6 X.

It is obvious that problem (P) coincides with problem (Po). To remain in the convex framework, we assume that $ is a convex function. Of course, when / : X —> R is convex we can find a function $ having the demanded property: $(x,y) := f(x) for all (x,y) E X xY. For

Perturbed

problems

107

obtaining useful results one chooses adequate perturbation functions, as we shall see in the sequel. Let $ : I x 7 4 l b e a convex function and h : Y -» E, h{y) = v(Py), its associated marginal function (see page 43); h is also called the value or performance function associated to problems (Py). As noted in Theorem 2.1.3(v), h is convex, while from Eq. (2.8) we have that dom/i = Pry(dom$). The problem (P)

min $(z,0), x £ X,

is called the primal problem; we associate to it, in a natural way, the following dual problem {D)

max (-$*(0,2/*)), y* € Y*.

It is obvious that (£>) is equivalent to the convex programming problem (!?')

min $*(0,2/*), y*

eY*.

The equivalence has to be understood in the sense that the problems (D) and (£>') have the same (e-)solutions; moreover v(D') = —v(D) (of course, for a maximization problem the notions of (e-)solution, local solution and value are defined dually to those for minimization problems). It is nice to observe that (£>') and (P) are of the same type. In the following results we establish some properties which connect the problems (P), (D) and the function h. Theorem 2.6.1 Let $ : X x Y -> E and h : Y ->• E be the marginal function associated to $ . Then: (i) h* (y*) = $* (0, y*) for every y*

eY*.

(ii) Let (x,y) € X x Y be such that ${x,y) 6 R. Then (0,j/*) € d$(x,y)

O h(y) = $(x,y)

and y* £ dh{y).

(hi) v(P) = h(0) and v(D) = /i**(0). Therefore v(P) > v(D); in this case we say that one has weak duality. (iv) Suppose that $ is proper, x € X and y* € Y*. Then (0,y*) € d$(x~, 0) «/ and only if x is a solution of problem (P), y* is a solution of (D) and v(P) = v{D) G E. Assume, moreover, that $ is convex. (v) /i(0) £ E and /i is Isc at 0 <£> u(P) = u(D) £ E; in this case one says that we have strong duality.

108

Convex analysis in locally convex spaces

(vi) h(0) G E and dh(0) / 0 & v(P) = v(D) G E and (D) has optimal solutions. In this situation S(D) = dh(0). (vii) Suppose that $ is proper. Then [ h is proper] •£> [ h* is proper] •£> [ h is minorized by an affine continuous functional] •& 3y* 6 7 ' , 3 a GE, V(x,y)eXxY Proof,

: $(x,y)

> (y,y*) + a.

(2.52)

(i) We have

h*(y*) = sup({y,y*) - h{y)) = sup ((y,y*) - inf y€Y

y€Y \

= sup sup((y,y*)

-$(x,y))

=

y€YxeX

$(x,y))

^ ^

sup

/

((x,0) + (y,y*) -

${x,y))

(x,y)€XxY

= **(<), y*). (ii) Assume that $(x,y) Theorem 2.4.2 (iii), h(y) < *(x,y)

G E. Let (0,j/*) G d$(x,y).

Then by (i) and

= (x,0) + (y,y*) - $*(0,i/*) = (y,y*) - h*(y*) < h(y),

and so h(y) = $(x,y) hold then

and y* G dh(y). Conversely, if these two conditions

*{x,y) = h(y) = {y,y*) - h*{y*) = (x,0) + (y,y*) - **(0,y'), whence, again by Theorem 2.4.2 (iii), (0,2/*) G d$(x,y). (iii) It is obvious that v(P) — /i(0); moreover, v(D) = sup (-$(0,r/*)) = sup ((0,2/*) -h*(y*)) y*EY*

= h**{0).

y*&Y*

Therefore v(P) >v(D). (iv) If (0,|7*) G<9$(x,0) then v(P) = h(Q) < $(x,0) = -$*(0,2/*) < v(D) = x**(0) < h(0), whence x is a solution of (P), y* is a solution of (£>) and v(P) = v(D) G E. Conversely, if these last assertions are true, we have -$*(0,j7*) = h**(0) = h(Q) = S(z,0) G E, whence ( z , 0 ) G d o m $ and §(x,0)+

**(0,y*) = (x,0) + (0,y*),

Perturbed problems

109

which shows that (0,y*) G d$(x,0). Assume now that $ is convex. (v) If h is lsc at 0 and h(0) G R we have that h(0) G E, whence h** = ~h. Therefore v(D) = h**(0) = ft(0) = v{P) G E. Conversely, if this last relation is true, then h(0) = h(0), i.e. h is lsc at 0. The conclusion holds. (vi) Suppose that /i(0) G E and dh(0) ^ 0; then /i is lsc at 0, whence, by (iv), we have that v(P) = v(D) G E. Let y* G dh(0). Then h(0) + h*(y*) = 0, whence Vy* G Y* : v(D) = v(P) = h(Q) = -h*(y*) = -**(<), IT) >

-*'{0,y*).

Therefore 0 ^ dh(0) C S(D). Conversely, suppose that v(P) = v(D) G R and that (D) has solutions. Let y* G S(D). Then ft(0) = h**(0) = -h*(y*) G E, whence y* G 9ft(0). Thus we have that 0 # S(D) C 0ft(O). (vii) The mentioned equivalences are obvious since h is convex, and dom/i^0. • We say that the problem (P) is normal if v(P) = v(D) G E; (P) is stable if w(P) = v(D) G R and (D) has optimal solutions. Theorem 2.6.1 above gives characterizations of these notions. In the following result we establish formulas for deh(y) and deh*(y*) when h is proper. Theorem 2.6.2 e G E+ . Then:

Let $ G A(X x Y) satisfy condition Eq. (2.52) and

(i) /or every y G dom h

deh(y) = f l , > 0 LUjcfo* e y * I (°'^) G ^+^0^)} 7)>0

*(x,y)
(ii) /or every y* G dom/i* ; cU*(y*) = p |

cl{yGY|3xGX

: (Q,y<) €

de+v*(x,y)}.

110

Convex analysis in locally convex spaces

Proof,

(i) It is obvious that

n

{»*i(o,!/ , )6fl t+ ,^»)}

n

V>0

$(x,y)
c

n.„U ( = >*l(o,i/*)ea e + ) J $( a :,i/)}

1

l

ri>0

^^x€X

C a e /i(y). Let y* G deh(y), i.e. h(y) + h*{y*) < (y,y*) +e, and let rj > 0 and a; € X be such that $(x,y) < h(y) + rj. Then $(x, 2/) + $*(0, y*) < ft(y) + r, + h*{y*) < (y, y") + e + r,, i.e. (0,2/*) € d£+n$(x,y).

Therefore

dsh(y) C f|

f|

{y*\ (0,y*) G &+„*(*, y)},

V>0
and so the desired equalities hold. (ii) Let y € d£h*(y*) and r\ > 0. Then />**(
= % ) + h*(y*) < (y,y*) + e.

It follows that for every V G N(j/) there exists yv G V such that Ml/v)+ &*(!/*) < + £ + ??• From the definition of h we get xy E X such that $(xv,yv)

+ h*(y*) = $(xv,yv)

i.e. (0,y*) G de+n
+ $*(0,y*) < (yv,y*)

+e + rj,

Therefore

j / G c l { t / G y | 3 a ; G X : (0,y*) G 3 £ + 7 ? $(z,y)}. Taking into account that 77 > 0 is arbitrary, we obtain that d£h*(y*)cf)

d{y\3xGX

: (0,y') G

de+r,*(x,y)}.

Let y be in the set from the right-hand side and 77 > 0. Then, using the continuity of y*, there exists Vo G N(y) such that (z,y*) < (y,y*) + r)/2 for every z G VQ. On the other hand, for every V G N(y) there exist

Perturbed problems

(xv,Vv) hence

111

€ X x Y such that yv GVTlVo and (0,y*) G

Kw) + h*{y*) < ${xv,yv)

+ $*(0,y*) < (yv,y*)+e

de+r,/2$(xv,yv);

+ i]/2 < (y,v*)+e+T].

Therefore V V e N f e ) : inf % ) + A*(y*)< (J/,!/*>+£ + »/, yev i.e. h(y) + h*(y*) < (y,y*) +E + TJ. Since TJ > 0 is arbitrary, we obtain that h"(y) + h*(y*) = h(y) + h*(y*) < (y,y*) + e, i.e.ytd£h*(y*). • The preceding result is useful for deriving formulas for e-subdifferentials for other functions. Theorem 2.6.3 Let $ E T(X x Y) and cp : X -» E, ip(x) := Then for every e E E+ and every x E dom ip we have dMx)=C\ = f|

F(x,0).

w*-c\{x*eX*\3y*eY*

: (x*,y*) E

d£+v$(x,0)}

w*-d{x*eX*\3y*eY*

: (x,0) E

de+v^(x*,y*)}.

Proof. Let us consider the spaces X*, Y* endowed with their weak* topologies and the function k:X*^W,

k(x*):=

inf

y*€Y*

$*(x*,y*).

Then k* : X ->• I , A;*(a;) = $**(x,0) = $(x,0) = ip(x). Using Theorem 2.6.2, we have dMx)

= dEk*(x) = f l = f| 1

The proof is complete.

w*-c\{x* | 3j/* : (a;,0) E &+-,**(**>!/*)} >J1>0 '77 X )

W*-C1{Z*|3T/*

: (x'.y'Jeft+^O)}. D

Let us apply the results of Theorems 2.6.2 and 2.6.3 in some particular situations. Stronger conclusions, but under more restrictive conditions, will be established in Section 2.8.

112

Convex analysis in locally convex spaces

Corollary 2.6.4 for which

Let A G H{X,Y)

3y* £Y*,

and f : X —> R be a convex function

3a G R, Vz e X : f(x)>

(x,A*y*)+a.

If (Af)(y) G 1 (O y G A(dom/) = d o m A / ) , j / * G dom(/* o A*) and e > 0, then

de{Af){y) = fl n>0

[J

A-\de+rif{x))

Ax=y

=n

n

»7>0

A*-\d£+rif(x)),

Ax=y,f(x)<(Af)(y)+r,

and ds(f*oA*)(y*) Proof.

= f]d{yeY\3xeX

: Ax = y, A*y* G

de+nf(x)}.

Let us consider K *,y) ,in

•=

' | oo

if

Axjty.

Then $*(x*,t/*) = /*(x* + A V ) and (O.y*) G 0„$(a;,y) <S> Ax = y, A*j/* G fy/fr). The result follows immediately from Theorem 2.6.2. Corollary 2.6.5

Let A G L(X, Y) and f G T(Y).

d£(foA)(x)

=

D Then

O^lv'-dA^de+rffiAx))

for every x G A _ 1 ( d o m / ) = dom(/ o A) and every e > 0. Proof. Let us consider $ : I x y ^ l , $(x,y) := f(Ax+y), and ip(x) :— $(x,0) = f(Ax). Then$*(x*,y*) = f(y*) ii A*y* =x*, $*(x*,y*) = oo otherwise. Moreover (x*,y*)edv*(x,0)&A*y*=x'

and f(Ax) + f*(y*)

& A*y* = x* and y* G The conclusion follows from Theorem 2.6.3.

< (Ax,y*) + rj

dvf(Ax). •

The fundamental

Corollary 2.6.6 3x* eX*, If (hnf2)(x)

duality

formula

113

Let / i , / 2 € A(X) for which

3a e l , \/xeX,

V i e {1,2} : fi{x) > {x,x*) + a.

e E and e > 0, then:

de(fiDf2){x)

= f) n>0

1J

(0 e i /i(a; -

y)ndeJ2(y))

yeX,Ei>0,e+n=ei+£2

=n n ri>0yeSrl(x)

u (^x/i^-^n^/ad/)),

£i>0,e+r)=£i+£2

d{huf2){x) = r\v>0Uy€X (Wx

- y)n ^/a(i/)).

where Sv(x) := {y e X \ Mx - y) + f2{y) < {fiDf2)(x)

+ r?}.

Proof. Let us consider / : X x X -)• E, f{x\,x2) := fi(xi) + f2(x2) and A e £ ( X x X , X ) , A(xi,x2) := £i + £2- The conclusion follows from Corollary 2.6.4. • Corollary 2.6.7 then:

Let / i , / 2 € T(X). / / a; e d o m / i n d o m / 2 and e > 0

ds{h+mi)=n ™*-d ( u (^Aw+aeaMx)) 7)>0

5(/i + f2)(x)

= fl

y£i>0,£+77=ei+e 2

t i T - d ^ / i C * ) + d„f2(x)).

Proof. Let us consider / : I x I - > I , / ( ^ i i x2) := / i ( x i ) + 72(^2) and A e £(X, X x X), Ax := (x,x). The conclusion follows from Corollary 2.6.5. •

2.7

The Fundamental Duality Formula

The following theorem is very useful for obtaining important results in convex programming; this is the reason for calling formula (2.53) the fundamental duality formula of convex analysis. Theorem 2.7.1 Let $ e A(X x Y) be such that 0 6 P r y ( d o m $ ) . Consider Yo := lin (Pry (dom $ ) ) . Suppose that one of the following conditions is satisfied:

114

Convex analysis in locally convex spaces

(i)

there exists Ao £ E such that VQ := {y £ Y \ 3x £ X, $(x,y)

<

AO}S:NVO(O);

(ii)

there exist Ao G E and x0 £ X such that VUeNx

: {y£Y\3x£x0

+ U, $(x,y)

< Ao} e Ky o (0);

(iii) there exists XQ £ X such that (a;o;0) £ d o m $ and $(xo,-) tinuous at 0;

is con-

(iv) X and Y are metrizable, epi $ satisfies condition (Hwz) on page 14 and 0£ i 6 ( P r y ( d o m $ ) ) ; 0e

(v) X is a Frechet space, Y is metrizable, $ is a li-convex function and i6 (Pry(dom$));

(vi) X is a Frechet space, $ is Isc and 0 £

j6

(Pry(dom$));

(vii) X, Y are Frechet spaces, $ is Isc and 0 €

lc

(Pry(dom$));

(viii) dimlo < 00 and 0 £ *(Pry(dom$)); (ix) there exists Xo £ X such that sets {0}, P r y ( d o m $ ) are united.

$(XQ,-)

is quasi-continuous and the

Then either h(0) = —00 or h(0) £ E and h\y0 is continuous at 0. In both cases we have inf $ ( x , 0 ) = max ( - **(0,i/*)). x€X

'

y*€Y'

v

"

(2.53) '

Furthermore, x £ X is a minimum point for $(-,0) if and only if there exists y* £ Y* such that (0,y*) £ d$(x,0). Proof. Since 0 £ P r y ( d o m $ ) , h(0) < 00. If h(0) = —00 we have h*(y*) — °° f° r every y* 6 Y*, whence —$*(0, j/*) = - 0 0 = h(0) for every y* £ Y*. Therefore the conclusion is true in this case. Let us consider now the case h(0) £ K. Suppose that condition (i) is verified. Then h\y0 is bounded above by Ao on VQ. Since h is convex, h\y0 is continuous at 0. By Theorem 2.4.12 we have that dh(0) f 0. Then relation (2.53) follows Theorem 2.6.1 (vi). (ii) =£- (i) This implication is obvious (just take U — X). (iii) => (ii) Suppose that (iii) holds. Since $(xo,-) is continuous at 0, the set V0 := {y | $(xo,y) < $(x0,0) + 1} is a neighborhood of 0 (in particular YQ = Y). Taking Ao := $(a;o,0) + 1, it is obvious that V0 C {y \3x £ x0 + U : $(x,y) < A0} for every U £ Nx- Therefore (ii) holds.

The fundamental

duality

formula

115

(iv) => (ii) Suppose that (iv) holds. Let us consider the relation 31 : X xR=$Y whose graph is given by

grtt := {(x,t,y) \ (x,y,t) € epi$}. From the hypothesis 51 satisfies condition (Hwi), whence, by Proposition 1.2.6(i), % satisfies condition (Hw(z,i)), too. Moreover 0 € i6 (Im3?). Let (xo,io) e X x R be such that 0 6 Jl(xo,to)- Applying Theorem 1.3.5 we obtain that "R{{x0 + U)x ] - oo,t0 + 1]) £ ^y 0 (0) for every U eNX- This shows that (ii) holds with Ao := t0 + 1. (v) => (ii) By Proposition 2.2.18, there exist a Frechet space Z and a cs-closed function F : Z xX xY -^R such that $(x, y) = inf z e z F(z, x, y) for all (x, y) € X xY. The conclusion follows like in the preceding case by replacing X by Z x X, x by (z,x) and U by Z x U. (vi) => (ii) The proof is the same as for (iv) =$• (ii) with the exception that one uses Ursescu's theorem (Theorem 1.3.7) instead of Simons' theorem. It is obvious that (vii) implies conditions (iv), (v) and (vi). If (viii) is verified, the conclusion follows from Theorem 2.4.12. (ix) => (i) It is obvious that $(a; 0 , •) > h. By Proposition 2.2.15 we have that h is quasi-continuous. It follows that rint(dom/i) ^ 0. Using Proposition 1.2.8 we obtain that 0 £ rint(dom/i). Therefore h\y0 is continuous at 0, and so (i) holds. Of course, if x is a minimum point for $(-,0) then x is a solution of (P) (from p. 107); it follows that $(x,0) = v(P) = v{D) G R. Let y* be a solution of (D) (in our conditions a solution exists). Then, from Theorem 2.6.1(iv), (0,2/*) £ d$(x,0). The converse implication follows from the same result. • When the properness condition in the preceding theorem is violated relation (2.53) is automatically verified. Indeed, in this case there exists (xi,yi) £ X x Y such that $(xi,j/i) = - o o , and so h(yi) = —oo. Because 0 € '(dornh), by Proposition 2.2.5 we have that h(0) — - o o . In applications it is important to have conditions on $ which also ensure that the functions $, $(a;, y) := <&(x,y) — {x,x*) with a;* 6 X*, satisfy them. Such conditions are conditions (ii)-(ix) of Theorem 2.7.1, but this is not always true for (i).

116

Convex analysis in locally convex spaces

Other conditions of this type are: 3A 0 G R, 3B G %x : {y G Y | 3x G B, $(x,y) V?7 G Kx, 3A > 0 : {yeY\3x£XU,

$(x,y)

< A0} G >JVo(0), (2.54) < A} G Ny o (0), (2.55)

where "Bx is the class of bounded subsets of X and, as in Theorem 2.7.1, Y0 = l i n ( P r y ( d o m $ ) ) . Proposition 2.7.2 Let $ G A(X x F ) . Conditions (i) — (viii) foez'ng those from Theorem 2.7.1, we have: (iii) =>• (2.54) =S> (2.55) «• (ii) => (i), (2.55) => (2.54) «/ X is a normed vector space, (vii) =>• (iv) A (v) A (vi), (iv) V (v) V (vi) =* (ii) and (viii) =* (2.54). Moreover, taking D = P r y ( d o m $ ) , one has: if dim(linZ)) < oo t/ien *D = rint£>; i/ X, Y are metrizable, e p i $ satisfies H(x) and lbD ^ 0 then thD = rintZ?; similarly for the situations corresponding to conditions (v)-(vii). Proof. The implications (vii) =>• (iv) A (v) A (vi), (iv) V (v) V (vi) =>• (ii) =*• (i) were already observed (or proved) during the proof of the preceding theorem. The implication (iii) =$• (2.54) is obvious; just take B — {XQ}. (2.54) =>• (2.55) Let U G K x ; there exists /x > 0 such that B C fill. Taking A = max{Ao,/i} we have that {yeY\3x£B,

$(x,y)

< A0} C {y G Y \ 3x G ^?7, $(a;,j/) < A0} C f e e Y | 3 x G XU, 9(x,y)

< A}.

The conclusion follows. (ii) =>• (2.55) Consider A0 G M. and zo G X given by (ii). Let U G N * . There exists fi > 0 such that a;0 € /if/. Let V = {y€Y\3x£x0 + U, $(x,y) < Ao} G Ny 0 . Taking A = max{Ao,/« + 1} and y € V, there exists x G x0 + U such that $(x,y) < A0. As x G x0 + U C /J,U + U = (fi + 1)U C At/, the conclusion follows. (2.55) =>• (ii) It is obvious that there exists x0 G X such that $(xo, 0) < oo. Consider Ao = max{$(a;o,0),0} + 1 and let U G Nx- There exists Uo G Nx such that UQ + Uo C U. There exists also Ai > 0 such that XQ G Aif/0. By hypothesis, there exists A > Ao + Ai such that Vb = {y G y | 3a; G AE/0, #(a;,i/) < A} G Ny o (0). Let V = X^VQ and take yeV. As Xy G Vo> there exists x' G Af/o such that $(x',Xy) < X. It follows that $ ((1 - A^Xzo.O) + A " V , A v ) ) < (l-A- 1 )$(a;o,0)+A- 1 $(a;',A 2 / ) < A0,

The fundamental

duality

formula

117

whence $ (x,y) < Ao, where x := xo + X~1(xl — Xo) G Xo + Uo + Uo C Xo + U. (2.55) => (2.54) when X is a normed vector space. Take U = Ux = {x G X | ||x|| < 1}. There exists A0 > 0 such that {y G Y \ 3x G X0U, ${x,y) < Ao} € Ny o (0). As XoU is bounded, the conclusion follows. (viii) =$• (2.54) Suppose that dim Yb < oo and 0 £ i ( P r y ( d o m $ ) ) . It follows that there exist j / i , . . . ,ym G Pry(dom$) such that Vo = co{y\,..., 2/m} £ !Nyo(0). For every i € l , m there exists Zj G X such that (xi,yi) G d o m $ . Let Ao = max{$(a;j,y;) | 1 < i < m} and B — co{xi,..., xm}. It is obvious that B is bounded and for y G Vo there exist A j , . . . , Am > 0 with Y!T=i ^i = 1 s u ch that j / = YlT=i ^'Vi- Then a; = YllLi ^ixi € -^ anc ^ *(z,2/) < A0. The fact that lD — v'mtD if dim(lin£>) < oo is obvious. Let X, Y be metrizable, e p i $ satisfy (Hx), and consider y0 G lbD. Taking <J?0 defined by
for every x* G X*. In particular ip G T*(X*). Proof. It is obvious that $(-,0) G r ( X ) in our conditions. Let x* G X* and consider $ : I x F - > I denned by $(x,y) := $(x,y) — (x,x*). As observed above, the function $ satisfies the same condition as $ among those mentioned in the statement of the corollary. Applying Theorem 2.7.1 (and eventually the preceding proposition), we have that inix&x $(x,Q) = m a x y . e y . ( - $ * ( 0 , ?/*)). But $*(0,2/*) = $*(x*,y*), and so the conclusion follows. • We state another duality formula which will be useful in the sequel. Theorem 2.7.4 Let F G A(X x Y), 6 : X =t Y be a convex multifunction, and D = U{G(x) — y \ (x,y) G d o m F } . Assume that 0 G D and let Y0 = linD. If one of the following conditions holds: (i) for every U G Nx there exist A > 0 and V G Ny0 such that {0} x V C g r C n (XU xY)-[F<

A];

Convex analysis in locally convex spaces

118

(ii) there exist Ao G K, B G T>x and VQ G Ny0 such that {0} x V0 C grC n (B x Y) - [F < A0]; (iii) there exists (xo,yo) G grC (~l d o m F such that F(xo, •) is continuous at y0; (iv) X, Y are metrizable, 0 6 lbD and either F is cs-complete and C is cs-closed, or F is cs-closed and C is cs-complete; (v) X, Y are Frechet spaces, F and C are li-convex, and 0 G lbD; (vi) dim YQ < oo and 0 € *£>, £/ien £/iere exists z* G Y* such that inf{F(x,y) Proof.

\ (x, y) G gr 6} = inf{F(x,y) + (z, z*) \ (x,y + z) e gr e } .

Let Z := Y and

* : ( I x Z ) x 7 4 l ,

$(x,z;y) := F(x,z) + t g r e ( x , y + z).

It follows easily that $ is convex and Pry(dom$) = D. It is obvious that the conclusion of the theorem is equivalent to inf( X ] 2 ) e xxZ^( a ; , 2 : iO) = max^.gy. — $*(0,0;2/*). So we have to show that if one of the conditions of the theorem is verified then a condition of Theorem 2.7.1 holds. If (i) holds it is immediate that $ verifies condition (i) of Theorem 2.7.1. The implication (ii) => (i) is obvious. We also have that (iii) => (ii); just take B := {x0}, A0 := F(x0,yo) + 1 and V0 = {y G Y \ F(x0,y0+y) < A0} (in this case Y0 — Y). Similar to the proof in Proposition 2.7.2 we have that (vi) =>• (ii). (iv) => (i) Let Y^n>i ^n(xn, zn, yn, tn) be a convex series with elements of epi $ such that J2n>i ^nXn and J2„>i ^nzn are Cauchy, J2n>i Kyn = y G Y and J2n>i ^tn = t G K Then (xn,zn,tn) G e p i F and (xn,zn + yn) G grC for every n > 1. If F is cs-complete it follows that J2n>1 Xnxn and S n > i ^nZn are convergent with sums x G X and z G Z, respectively; moreover, (x,z,t) G epi.F, whence (x,z + y) G grC since grC is cs-closed. The same conclusion holds in the other case. Therefore $ verifies condition (H(x,z)) in our hypotheses. Hence condition (iv) of Theorem 2.7.1 is verified. So, by Proposition 2.7.2, relation (2.55) holds. Therefore for U x Y G Nxxz there exists A > 0 such that {y G Y | 3 (x, z) G XU x Y, $(x, z; y) < A} = {y&Y\3xG\U,

3zGZ

: (x,y + z) G e, F{x,z)

< A} G Ny o (0),

The fundamental

duality

formula

119

and so (i) holds. (v) => (i) Consider the set A:={(x,z,y)

eX

x Z xY

\y + z£

= {{x,y' -y,y)\{x,y')egre, = grC x {0} + {0} x {(-y,y)

G(x)} yeY}

\y

eY}.

Using Propositions 1.2.4 (ii) and 1.2.5 (ii) we obtain that A is li-convex (as sum of two li-convex subsets of a Frechet space). Since $(x,z;y) — F(x, z) + LA{X, Z, y) and the functions F and LA a r e li-convex, $ is li-convex, too. Thus $ satisfies condition (v) of Theorem 2.7.1; the other conditions being obviously satisfied, as in (iv) =4- (i), we get that (i) holds, too. • Note that every condition of the preceding theorem is verified by F, F(x,y) — F(x,y) — (x,x*), where x* € X*, when the same condition is verified by F. Taking gr 6 = X x {0}, the conclusion of the preceding theorem is just the conclusion of Theorem 2.7.1. Conditions (i), (ii), (iii) and (vi) become conditions (2.55), (2.54), (iii) and (viii) of Theorem 2.7.1, respectively; to conditions (iv) and (v) correspond slightly stronger forms of conditions (iv) and (v) of Theorem 2.7.1, respectively. Remark 2.7.1 If F(x,y) = f{x) + g(y) with / € A(X), g € A(Y), for condition (i) of Theorem 2.7.4 it is sufficient (and necessary if f,g have proper conjugates) to have VUelSx,

3 A > 0 , 3V € X y 0 : V C [g < A] - e(\U n [/ < A]),

while for condition (ii) of Theorem 2.7.4 it is sufficient (and necessary if / , g have proper conjugates) to have 3 A0 S 1, Be Sjr, V0 £ NYo

: V0 C [g < A0] - G{B n [/ < A0]).

Of course, these two conditions are equivalent if X is a normed space. Corollary 2.7.5 Let f € A(X) and A 6 &(X,Y). the following conditions is verified:

Suppose that one of

(i) / is continuous on int(dom/), assumed to be nonempty, and A is relatively open, i.e. A(D) is open in ImA for every open subset D C X; (ii) X and Y are metrizable, either (a) / is cs-complete or (b) / is cs-closed and gr A satisfies condition (Hx), and * 6 A(dom/) ^ 0;

120

Convex analysis in locally convex spaces

(iii) X is a Frechet space, Y is metrizable, f is a li-convex function and A(domf)jt
ib

(iv) X is a Frechet space, f is lower semicontinuous and %bA(dom f) ^ 0; (v) dim (lin A(dom/)) < oo. Then, either Af is —oo on *(^4(dom/)) or Af is proper and {Af)\y0 continuous on ' (A(dom f)). Moreover Vj/G^dom/)) Proof.

: (Af)(y) =max{{y,y*)

- r(A*y*)\y*

is

eY*}.

Let us consider y0 6 l (A(dom/)) and $ : X x Y -*• I ,

$(ar,y) := f(x) + tgTA(x, 2/o + y).

We have that Pry (dom $) = yl(dom/) - y0- If condition (ii), (iii), (iv) or (v) is verified, then $ satisfies condition (iv), (v), (vi) or (viii) of Theorem 2.7.1, respectively. Suppose that condition (i) is satisfied. (Obviously, it is impossible that condition (iii) of Theorem 2.7.1 be verified in this case: F(xo, •) = i{Ax0}-) Since A is relatively open we have that *(A(dom/)) = intim/i (A(dom/)) = A(int(dom/)) (Exercise!), whence j/o = AXQ for some Xo € int(dom/). It follows that / is bounded above on a neighborhood Vo of Xo, and so h (the marginal function associated to $) is bounded above by the same constant on J4(VO) — yo, which is a neighborhood of 0. Therefore condition (i) of Theorem 2.7.1 is verified. So, under each of the five conditions, we have that (Af)(y0)

= inf $(a:,0) = max (-$*(0,y*)) = max « y 0 , y * > - / * ( > l V ) ) , xex y*eY* yer*

doing similar calculations to those from Theorem 2.3.1(ix).

•

Corollary 2.7.6 Let f1}..., fn G A(X) and take f :- / i D • • • • / „ . Suppose that one of the following conditions is verified: (i) / i is continuous at some point in dom / i , (ii) i6

X is a Frechet space, the functions / i , . . . , / n are li-convex and

(dom/)^0, (iii) dimX < oo.

Then either f is —oo on ! ( d o m / ) or /|aff(dom/) is finite and continuous on (domf), in which case f is sub differentiable on this set. Furthermore, for every x € ' ( d o m / ) we have that

t

f(x) = max ((*, x*) - / • ( * • )

/ • ( * ' ) ) = ( / ; + ••• +

rn)*(x).

Formulas for conjugates and e-sub differentials

121

Proof. Recall that dom / = dom / i -\ + dom / „ . In the cases (ii) and (iii) the result is an immediate consequence of Corollary 2.7.5 taking $ and A as in Corollary 2.4.7, while for (i) one does the proof by induction. •

2.8

Formulas for Conjugates and e-Subdifferentials, Duality Relations and Optimality Conditions

In the preceding sections we have considered only situations in which it was simple to compute conjugates and e-subdifferentials: sum of functions with separated variables, convolution of convex functions, and functions of type Af. For the other types of functions, generally, it is more difficult to compute the conjugate functions or the e-subdifferentials. In this section, we intend to establish sufficient conditions, as general as possible, in order to ensure the validity of such formulas. In this section the spaces X,Xi,..., Xn and Y are separated locally convex spaces if not stated explicitly otherwise. We begin with the following result. T h e o r e m 2.8.1 Let F £ A(X x Y), A £ H{X,Y) and

E, x such that {0} x Vo C {{x,Ax)

\xeB}-[F<

Ao];

(ii) for every U £ Nx there exist A > 0 and V £ Ny0 such that {0} x V C {(x, Ax)\xe

At/} ~[F<

A];

(iii) there exists xo £ X such that (:ro,j4a;o) £ d o m F and F(xo,-) continuous at AXQ;

is

(iv) X and Y are metrizable, 0 E tbD and either F is cs-closed and gr A is cs-complete or F is cs-complete; (v) X is a Frechet space, Y is metrizable, F is li-convex and 0 £ lhD; (vi) X is a Frechet space, F is Isc and 0 £ lbD; (vii) X and Y are Frechet spaces, F is Isc and 0 £ %CD; (viii) dimy 0 < oo and 0 £ 'D,

Convex analysis in locally convex spaces

122

(ix) there exists XQ £ X such that F(xo, •) is quasi-continuous and {0} and D are unitedThen for x* £ X*, x £ domip and e > 0 we have: tp*(x*) = mm{F*(x* - A*y\y*) dM*) Proof.

= {A*y*+x*

\ y* £ Y*},

(2.56)

| (x*,y*) € dEF(x, Ax)}.

(2.57)

It is easy to verify that Vx* £ X* :
- A*y*,y*) \ y* £ Y*}

for every function F and every operator A £ &(X, Y). Consider $ : X x Y -» E, $(x,y) = F(x,Ax — y). Then $ satisfies one of the conditions (2.54) or (ii) - (ix) of Theorem 2.7.1 when F satisfies one of the conditions (i) — (ix), respectively. It follows that the function $, defined by
- (x,x*) \ x £ X} = inf {$(z,0) | x £

X).

Applying Theorem 2.7.1, we obtain that - V ( a ; * ) = i n f { S ( x , 0 ) | X e X} = max { - * ' ( 0 , -y") \y* £Y*}.

(2.58)

But **(0, -V*) = sup{(ar, a;*) + (y, -y*) - F(x, Ax - y) \ (x,y) £ X x Y} = sup{(a;,a;*) + (z - Ax,y*) - F(x,z)

\ (x,z) £ X x Y}

= sup{(a;, x* - A*y*) + (z, y*) - F(x, z) \(x,z)£Xx =

Y}

F*(x*-A*y*,y*).

Prom Eq. (2.58) we get immediately Eq. (2.56). Note that the inclusion "D" in Eq. (2.57) is true for every function F and every operator A £ L(X,Y). Let x £ domip and x* £ detp(x). Then (see Theorem 2.4.2) cp(x) + ip*(x*) <

(x,x*)+e.

By Eq. (2.56) there exists y* £ Y* such that
+ F*(x* - A*y\y*)

A*y*,y*),

< (x,x* - A*y*) + (Ax,y*) + e,

Formulas for conjugates and

s-subdifferentials

123

i.e. (x* - A*y*,y*) =: (x*,y*) £ deF(x,Ax). Therefore x* = x* + A*y*, which proves that the inclusion "C" in Eq. (2.57) holds, too. • Note that (viii) V (hi) => (i) => (ii), (iv) V (v) V (vi) =>• (ii), (vii) => (iv) A (v) A (vi), and (ii) =$• (i) if X is a normed space. Note also that similar properties to those stated in the second part of Proposition 2.7.2 can be given for the situations of Theorem 2.8.1. Corollary 2.8.2

Under the conditions of Theorem 2.8.1 we have that

inf F(x,Ax) xeX

= max ( - F*(-A*y*,y*)).

(2.59)

y*€Y*

Furthermore, x~ is minimum point for

there exist Xo £ R, B £ "Bx and VQ £ Ny0 such that V0CA{[f<\0\nB)-[g<\0\;

(ii) for every U £ J^x there exist A > 0 and V £ Ny0 such that VcA([f<X]nXU)-[g<X\; (hi) there exists xo £ d o m / n A~l(domg) AxQ;

such that g is continuous at

(iv) X,Y are metrizable, 0 £ lb(A(dom f) — domg), f and g have proper conjugates, either f, g are cs-closed and gr A is cs-complete, or f, g are cscomplete;

124

Convex analysis in locally convex spaces

(v) X is a Frechet space, Y is metrizable, f, g are li-convex functions and Oeib (A(dom / ) - dom g); (vi) X is a Frechet space, f,g are Isc and 0 £ lb(A(dom f) — domg); (vii) X, Y are Frechet spaces, f,g are Isc and 0 £ tc (^4(dom/) — domg); (viii) dimlo < oo and 0 € *(A(domf) — domg); (ix) g is quasi-continuous and A(dom/) and domg are united; (x) Y = Wl, qri(dom/) ^ 0 and 4 ( q r i ( d o m / ) ) l~l Momg ^ 0. Then for every x* 6 X*, x 6 domy? and e > 0 we have: p ' O O = min{/*(** - A*y*) + g*(y*) | y* € F * } , de
+ d£2g(Ax) \ si,e2 > 0, ei + e 2 = e},

^ ( x ) = fl/(a:) + A*(dg(Ax)).

(2.60)

P r o o / . We apply Theorem 2.8.1 to A and F : X x Y -> 1 defined by F(x,y) := f(x) + g(y). It is easy to see that if one of the conditions (i)(ix) holds, then the corresponding condition of Theorem 2.8.1 is verified. If (x) holds, using the properties of the algebraic relative interior in finite dimensional spaces (p. 3) and Proposition 1.2.7, we have *(i4(dom/)-domff) = i ( A ( d o m / ) ) - i ( d o m g ) = ^ ( q r i ( d o m / ) ) - i ( d o m g ) , and so (viii) holds, too. The conclusion follows then applying Theorems 2.8.1, 2.3.1 (viii) and Corollary 2.4.5. • Note that, as in Theorem 2.8.1, we have that (viii) V (iii) => (i) => (ii), (iv) V (v) V (vi) =» (ii), (vii) =>• (iv) A (v) A (vi), (ii) => (i) if X is a normed space, and of course, as mentioned in the proof, (ix) =>• (viii). Applying the preceding result we obtain a formula for normal cones. Corollary 2.8.4 Let A 6 Z(X,Y) and L C X, M C Y be convex sets. Suppose that one of the following conditions is verified: (i) there exists XQ S L such that Ax0 6 int M, (ii) X,Y M),

are of Frechet spaces, L,M

are li-convex and 0 € lb(A(L) —

(iii) d i m F < o o and 0 £ 1{A(L) - M). Then for every x € LC\ A~1(M) N(Lr\A-l(M);x)

= N{L;x) + A*{N(M; Ax)).

(2.61)

Formulas for conjugates and

e-subdifferentials

125

Proof. Using the preceding theorem for f := IL, 9 •= IM and A, formula (2.61) follows from formula (2.60). • Corollary 2.8.5 Under the conditions of Theorem 2.8.3 we have the following relation, called the Fenchel-Rockafellar duality formula, inf. {f(x)+g(Ax)) igA

= ma* ( - / ' ( - A Y ) y

g*(y*)).

€'

Furthermore, x is a minimum point for f + go A if and only if there exists y* G Y* such that -A*y* £ df(x) andy* £ dg(Ax). Proof.

We proceed as in Corollary 2.8.2 (or apply this corollary).

•

Two particular cases of Theorem 2.8.3 are important in applications: / = 0 and A = Idx- The next theorem is stated even for A replaced by a convex process C. T h e o r e m 2.8.6 Let g £ A(F) and C : X =4 Y be a convex process. Assume that 0 G D, where D := ImC — domg. Consider YQ := linD and the function ip : X —> R, 0 and V € Ky0 such that V C [g < A] - e(XU); (ii) there exist XQ € R, B € 3x X] - G(B);

and V0 € Ny0 such that V0 C [g <

(iii) there exists j/o € dorngfllmC such that g is continuous at yo; (iv) X, Y are metrizable, g has proper conjugate, either g is cs-closed and C is cs-complete, or g is cs-complete and C verifies (Hz), and 0 £ lbD; (v) X, Y are Frechet spaces, g, C are li-convex and 0 6 lbD, (vi) dim Y0 < oo and 0 6 ' D . Then Vi'el*

: ip*{x*)=mm{g*{y*)

\ x* G C*(
Moreover, ifxE domtp = C - 1 (domg) is such that 0 then de(p(x) C C* {deg(y)) (with equality i/grC is a linear sub space). Proof. Then

Let x* £ X*. Consider F : X x Y -)• I , F(x,y)


-
= -mi{F{x,y)

:= g(y) -

(x,x*).

\ (x,y) G grC}.

Convex analysis in locally convex spaces

126

If one of the conditions (i)-(vi) holds then the corresponding condition of Theorem 2.7.4 holds. (In fact, in case (iv), if g is cs-complete and C satisfies (Ha;) one verifies directly that the function $ from the proof of Theorem 2.7.4 satisfies (E(x,z)).) So, by Theorem 2.7.4, there exists y* £ Y* such that
+ (z,y*) | (x,y + z) £ grC}

= sup{(z, a;*) - g(y) - (y1 - y, y*) | (x, y') £ gr C, y £ Y) = sup{(y,y*) -g{y)

\ y £ Y}

+ sup{(a;,a;*) + (z,-y*)

- igre(x,z)

\ x € X,z € Y}

= 9*(v*) + t-(gr e)+ (x*, -y") = g*{y") + tgr e* (y*,x*). Since V / e T

: V*(x*)
+ Lgve.(z*,x*),

(2.62)

the conclusion follows. Let x £ domy = A~1(domg) be such that 0. Let x* € dE(p(x). Since • (ii) =» (i) and (iv) V (v) => (i). Important situations when gr C is a linear subspace are: C = A"1 with A £ &(X, Y) and 6 = A with A a densely defined closed operator {i.e. A : D(A) -> Y, D(A) being a dense linear subspace of X, A being a linear operator and gr A a closed subset of X x Y; see also Exercise 1.11). T h e o r e m 2.8.7 Let / , 0 and V £ N.*-,, VC[f<X]n\U-[g<

A];

S«C/J

that

Formulas for conjugates and

s-subdifferentials

127

(iii) there exists XQ G dom / n dom g such that g is continuous at XQ ; (iv) X is metrizable, f,g have proper conjugates, f is cs-closed, g is cs-complete and 0 G l 6 ( d o m / - domg); (v) X is a Frechet space, f,g are li-convex and 0 G l i > (dom/ — domg); (vi) X is a Frechet space, f,g are Isc and 0 € l 6 ( d o m / — domg); (vii) X is a Frechet space, f,g are Isc and 0 G I C (dom/ — domg); (viii) dimX 0 < oo and 0 G *(dom/ — domg); (ix) g is quasi-continuous and d o m / and domg are united; (x) X is a Frechet space, f,g are li-convex and (0,0) G lb[{(x,x) X} — dom / x dom g).

\x G

Then for x* € X*, x G dom / Pi domg and e > 0 we have: ( / +
+ dE2g(x) \ei,e2

> 0, ex + e2 =e),

d(f + g)(x) = df(x) + dg(x). Proof. Taking A = Idx, the conclusion follows from Theorem 2.8.3 under conditions (i) - (iii), (v) - (ix). If (iv) holds, taking Y = X and $(x,y) := f(x) + g(x — y), condition (iv) of Theorem 2.7.1 is verified. If (x) holds consider F : X x X -)• I , F{x,y) := f{x) + g{y) and A : X -> X x X, A(x) :— (a;, a;). Then condition (v) of Theorem 2.8.6 holds; applying it we obtain the conclusion, taking into account that A*{x*,y*) = x* + y*. • The same implications as in Theorem 2.7.1 (mentioned in Proposition 2.7.2) hold. Corollary 2.8.8 Let X be a Frechet space and f, g G A(X). / / / and g are li-convex then for every x G * 6 (dom/ + domg) we have (/Dg)(x) = (/*+$*)•(*) = m&x{(x,x*)

- f*(x*)

- g*(x*) \ x* € X*}. (2.64)

Proof. Let a;o G s 6 (dom/ + domg) and consider h G A(X), h(x) := g(a;o — x). Then dom/i = XQ — domg, and so d o m / - dom/i = d o m / + domg — XQ- Therefore 0 G l 6 ( d o m / — dom/i), and so condition (v) of

Convex analysis in locally convex spaces

128

Theorem 2.8.7 is satisfied. From formula (2.63) applied for x* = 0 we get (fn9)(x0)

= inf. (/Or) + h{x)) = - ( / + h)*(0) = - min ( / • ( * * ) + >»*(-**))• X*£X *

But h*{—x*) = s u p { - (x,x*) — g(xo — x) \ x € X} = g*(x*) — (x0,x*), and so (2.64) holds for x = x0. O Of course, one can obtain the conclusion of the preceding corollary also for other situations corresponding to conditions of Theorem 2.8.7. For example, if there exist A0 £ M, B € "S>x, VQ € Nx0 (x) s u c n that Vo C [/ < A0] n B + [g < A0], where X0 - aff(dom/ + dom^), then Eq. (2.64) holds. Proposition 2.8.9 Let f,g € A(X) and take D = d o m / - domg. If dim(lin£>) < oo then {D = rintZ?, while if X is metrizahle, f,g have proper conjugates, f is cs-closed, g is cs-complete and tbD ^ 0 then lbD = rint D; similarly for the situations corresponding to conditions (v)-(vii) of Theorem 2.8.7. Suppose that X is a Banach space and f, g are li-convex. Then for every x € lbD there exist n, A > 0 such that (x + nUx) n aff D C [f < A] n \UX - [g < A] D \UXProof.

$(x,y)

(2.65)

The first part follows from Proposition 2.7.2 taking Y = X and

=f(x)+g(x-y).

Suppose that X is a Banach space and / , g are li-convex; consider x £ lb D. Replacing / by / , f(u) — f(u + x), we may suppose that x = 0. It follows that condition (v) of Theorem 2.8.7 holds, and, as noted after its proof, condition (i) is verified. Therefore there exist 77 > 0, Ao G B. and B £"Bx such that rjUx n aff D C [/ < Ao] n B - [g < A0]. Taking A' > max{Ao, 0} such that B C X'Ux and A = A' + n we obtain that rjUx n aff D C [/ < A0] 0 B - [g < A0] D (B + nUx)

c [/ < A]n xux - [f < A]n \ux. The proof is complete. Another important result is the following.

•

Formulas for conjugates and

E-subdifferentials

129

Theorem 2.8.10 Let Y be ordered by the convex cone Q, f £ A(X), H : X —> Y' be convex and g £ A(y) be Q-increasing on H(domH) + Q. Then tp :— f + goH is convex. Assume that 0 £ D, where D :— H(domHC\ d o m / ) — domg + Q, and consider Yo := linD. Assume that one of the following conditions holds: (i) for every U £ J^x there exist A > 0 and V € Ky0 such that V C H (XU n [/ < A] n d o m F ) - [g < A] + Q; (ii) there exist XQ 6 R, B £ 1$x and VQ £ Ny0 such that V0 C H (B n [/ < A0] n domH) -[g<

A0] + Q;

(iii) there exists XQ £ d o m / fl i f - 1 (domg) such that g is continuous at H(x0); (iv) X, Y are metrizable, f, g have proper conjugates, either f, g are cs-closed and epiif is cs-complete, or f,g are cs-complete and epiH is cs-closed, and 0 G lbD; (v) X, Y are Frechet spaces, f,g, epiH are li-convex and 0 £ (vi) dim Y0 < oo and

tb

D;

i

0£ D;

(vii) g is quasi-continuous, and domg and H(domH united.

f~l d o m / ) + Q are

Then for x* £ X*, x £ dormp = d o m / n H~1(domg) have:

and e >Q, we


deV{x) = | J {0Er (f + y*° H){x) \y*£Q n

dE2g(H(x)),

(2.66) El

+ e2 = e} . (2.67)

Proof. have

First observe that for all x* £ X*, y* £ Q+ and x £ dom<£ we

V ' O O < (/ + V* o H)*(x*) + g*(y*), de
+

+ y*o H)(x) \y*£Q n

de2g(H(x)),

(2-68) El

+ e2 = e], (2.69)

Convex analysis in locally convex spaces

130

without any supplementary condition on / , g and H. Indeed, let x € dom<^ = d o m / n H-^domg), x* € X* and y* £ Q+. Then f{x) + (y* o H)(x) + (f + y*o H)* (x*) > (x, x*), g(H(x))+g*(y*)>(H(x),y*), whence, adding them side by side, we get g*(y*) + (/ + y* ° H)* (x*) > (a;, a;*) - 0, £ = £x + e 2 . Then f{x) + (y* o H){x) + (f + y*o H)* {x*) < (x, x*) + ex, g(H(x))+g*(y*)<(H(x),y*)+e2. Adding them side by side we get ip(x) + g*(y*) + (/ + y* ° H)* (x*) < {x,x*) + e. Using Eq. (2.68) we obtain that x* € deip(x). Therefore Eq. (2.69) holds. Let F(x, y) := f(x) + g(y) and G : X =t Y with gr C := epi H; then F £ A(X x Y) and 6 is convex. Since g is increasing on H(dom H) + Q, it follows that (p(x) = int{F(x,y) + iep\H{x-,y) \ y 6 Y] for every x € X. Hence ip is the marginal function associated to the convex function F + iepi H ', hence ip is convex. If one of the conditions (i) — (vi) is verified, then F and 6 satisfy the corresponding condition of Theorem 2.7.4. (When / and g are cs-complete and have proper conjugates, similarly to the proof of Proposition 2.2.17, we get that F is cs-complete.) As noticed after the proof of that theorem, also the perturbed function F, F(x,y) = F(x,y) — {x,x*), satisfies the same condition. Therefore there exists z* £ Y* such that a : =

jnf

,

=

.1TUix)+g{y)-{x,x''))

inf

(f(x)+g(y)-(x,x*)

+

(z,z*))=:(3.

{x,y+z)EepiH

Using again the fact that g is increasing on H(domH) a =

Jnf

w

xedom H

=

inf iGdom H

j?^

n

+ Q, we have

VW + 9{y) ~ (x,x*))

yeH(x)+Q

(f(x)+g(H(x))

- {x,x*)) = ~
Formulas for conjugates and

s-subdifferentials

131

and P=

inf

inf

(f(x) + g(y) - (x, x*) + (H(x) +q-y,

z")).

iGdomif q€Q, y€Y

It follows that P = - o o if z* $ Q+. If z* £ Q+ then -0=

sup x€dom

sup((x,x*)-f(x)-(H(x),z*)

+

(y,z*)-g(y))

Hy£Y

= (/ + Z*o #)*(*•) + • K, F(x,y) := f(x)+g(H(x)+y)(x,x*). Because 0 € D and g is Q-increasing on H(domH) + Q, there exists xo € d o m / n H~1(domg). It follows that F(x0, •) is quasi-continuous and {0} and D — Pry (dom F) are united. Applying Theorem 2.7.1(ix) we have that inf x e x F(x,0) = max y . e y.(—F*(0,y*)). With a similar calculation as above we obtain that Eq. (2.66) holds. Let x 6 donup and x* 6 d£ip(x). Using Eq. (2.66), there exists y* € Q+ such that ip*(x*) = (f + y* o H)*(x*) + g*{y*). It follows that {f+y*oH){x)+{f+y*oHY{x*)-(x,x*)+9{H{x))+g*{y*)-{H{x),y*)

< e.

Taking £ l := (/ + y* o tf)(x) + (/ + y* o #)*(x*) - (x,x*) (> 0 by the Young-Fenchel inequality) and e 2 = e - ei, we have that y* € d£2g(H(x)) and Z* € d£l(f + y* oH)(x). Therefore the inclusion C holds in Eq. (2.67); taking into account Eq. (2.69), we have the desired equality. D We use the preceding theorem to obtain formulas for conjugates and subdifferentials of functions of "max" type. Corollary 2.8.11

Let / i , . . . , / „ E \(X)

y.X^W,

and

¥>(*):=/i(z)V---V/„(z).

Suppose that dom ip = f}™=1 dom /i ^ 0 and e € IR+. For x € dom ip denote by I(x) the set {i G l , n | fi(x) = p*{x*) = min{(Ai/i + • • • + A n / n )*(x*) | ( A i , . . . , An) e A„},

Convex analysis in locally convex spaces

132

while for every x £ dom
(Ai, . . . , A„) G A„,

77 e [0,e], Yli=1Xi^x^

- ¥>(x)+ri-£o}

,

dV{x) = U { 5 (E" = 1 A J i ) {x) I (Al!' • •'An) € A"' Vifl{x) Proof.

: Ai = o}.

Let us consider the functions

H:X->(W\Rl),

H(x):=( "•" 3: «"-•«,

(A(*)>••"'/«(*)) [00

if x € fX=i dom/,, otherwise,

g(y):=yiV.---Vyn.

It is clear that condition (iii) of the preceding theorem is verified. Taking into account the expressions of g* and deg(y) given in Corollary 2.4.17 we obtain immediately the relations from our statement. D An application of the previous result is given in the following example. Example 2.8.1

Let / € A(X) and consider /+ := / V 0. Then

f df(x) df+(x) = l U { W ) ( x ) | A e [ 0 , l ] } { d(0f)(x) = didomf(x)

if f(x) > 0, if f(x) = 0, if f(x)<0.

A useful particular case of the result in Corollary 2.8.11, in which we can give explicit formulas for ip* and detp without supplementary conditions, is presented in the following corollary. Corollary 2.8.12
--1J=I

Let f{ 6 A(X,) for i € T~n and Xi-+W,

tp(x1,...,xn):=f1(xi)V---Vfn(xn).

For x = (xi,... ,xn) S domtp = Yl7=i dom/j consider I(x) := {i G l , n | fi(Xi) =
(Xu...,\n)eAn}.

(2.70)

Formulas for conjugates and e-subdifferentials

For x — (xi,..., dMx)

133

xn) E dom

= {j{~[[n.=1dei(^fiKxi)\

Eni=o

( A i , . . . , A „ ) G A n , et > 0, V~^n £i = £

1 Xi Xi

' Z^i=1 ^ ^

- ^

£

- o) >

S ^ ^ - l J l n ^ ^ A i / O C ^ ) ! (A1,...,An)€A„, VigJ(a:) : A; = o} . Proof. We apply the preceding corollary to the functions fa : X ->• R, /i(a;) := fi(xi), then we use Theorem 2.3.1 (viii) for arbitrary n and Corollary 2.4.5. • Other situations when one has explicit formulas for tp* and d
3
+ d(»g)(x) I (A,M) G A 2 , A / ( X ) + ng(x) =
Proof. When / and g verify one of the conditions (i)-(iii), (v), (viii)— (x) of Theorem 2.8.7 and A,/x > 0, then the functions A/ and fig also verify the same condition. If condition (vi) or (vii) holds then, by the relations among the classes of convex functions on page 68, condition (v) holds. So, (Xf + ng)*{x*)=Toia{(Xf)*{u*) + (jjig)*{v*) \u*+v* = x * } a n d d(\f + ng)(x) = d(Xf)(x) + d(fig)(x). Using Corollary 2.8.11 one obtains the conclusion. • Taking into account that for x G dom / , one has df(x) + N(dom f, x) = df(x) (see also Exercise 2.23), d(Xf)(x) = \df(x) for A > 0 and d{0f)(x) = N(domf,x), we get for f,g G A(X) and x G X with f(x) = g(x) G R,

U{W)W+8WWI(A,|f)6A 2 } -co(df(x)Udg{x))

+ N(domf,x)

+N(domg,x).

This formula can be extended easily to a finite number of functions. Using Corollary 2.8.12 we get the following result concerning the maxconvolution.

134

Convex analysis in locally convex spaces

Corollary 2.8.14 have that

Let / i , / 2 € A{X) and e £ 1+ . For every x* £ X* we

(hQf2y(x*)=rnin{(\1f1y(x*)

+ (\2f2y(x*)

| (A1;A2) £ A 2 } .

(2.71)

Suppose that (/i0/ 2 )(a:) = max{/i(2;i),/ 2 (a; 2 )}, where xi £ d o m / i , x~2 £ dom /2 and x = x~i + x2. Then

de{hOh){x) = \J{dEl(^h)(xi)ndE2(\2f2)(x2) e i , e 2 > 0 , £i+e2<e

+ X1f1(x1)

I (Ai,A2) e A2,

+ X2f2(x2)-(f10f2)(x)}.

(2.72)

Furthermore, if fi(x\) = f2(x2) *^en Eq. (2.73) is verified, while if fi(xi) 72(^2) *^en i?g. (2.74) is verified, where d(fi0f2)(x)=[j{d(X1f1)(x1)nd(X2f2)(x2) d(fi0f2)(x) Proof.

I (A1;A2) G A 2 } ,

= dMxx) nN(domf2-:x2).

>

(2.73) (2.74)

Let x* £ X*; then

{fi0f2)*(x*)

= s u p i e X ( ( x , x * ) - i n f f / ^ ) V / 2 ( z 2 ) I *! + x2 = x}) = sup{(ii,a;*) + (a;2,a;*) - fi(xi)

V f2(x2)

\x1} x2 £ X)

= min{(A 1 / 1 )*(x*) + (A 2 / 2 )*(s*) | (Ai,A2) £ A 2 } , the last equality being obtained by using Eq. (2.70). The inclusion "D" of Eq. (2.72) can be verified easily. Consider x* £ de(fi0f2)(x). By Eq. (2.71), there exist (Ai, A2) £ A 2 such that (/i0/2)*(x*) = (Ai/i)*0O + (A 2 / 2 )*(x').

(2.75)

Using the preceding relation we obtain that 0 < [ ( A 1 / 1 ) ( ^ 1 ) + (A 1 / 1 )*( 2; *)-(5; 1 ,a;*)] + [(A 2 / 2 )(i 2 ) + (A 2 / 2 )*(:c')-<S 2 ) a :*)] < (fiOh)(x)

+ ( / i O / 2 ) * 0 0 - (x,x*) < e.

Taking e, := (Aj/i)(zj) + {Xifi)*(x*) - (x~i,x*) > 0, i £ {1,2}, using the preceding relation and Eq. (2.75), we get ( / i 0 / 2 ) ( z ) + £ 1 - Ai/i(3fi) + e 2 - A 2 / 2 (x 2 ) < e, and so x* belongs to the set on the right-hand side of relation (2.72).

Formulas for conjugates and

s-subdifferentials

135

Let us prove now the equalities (2.73) and (2.74). The inclusions "D" follow directly from Eq. (2.72). Let us prove the converse inclusions. Let x* G d{faOfa)(x) = d0(faOfa)(x). By Eq. (2.72), there exist (Ai, A2) G A 2 and £i, £2 > 0 such that x* £1 + e 2 < Xifi(xi)

£dei(Xifi)(x1)ndea(Xif2)(x2), + A 2 / 2 (x 2 ) - (faOfa){x) < 0.

Therefore £1 = £2 = 0 and Ai/i(xi) + A 2 /2(x 2 ) = {fa(>fa){x); hence x* belongs to the set on the right-hand side of Eq. (2.73) when / i ( x i ) = fa{x2)If fi{x~i) > / 2 ( ^ 2 ) , the preceding relation shows that A2 = 0 , Ai = 1, and so x* G 9/i(11),

x* G 0(0 • / 2 )(z 2 ) = <9idom/2(^2) = -/V(dom/ 2 ;x 2 ).

This shows that x* belongs to the set on the right-hand side of relation (2.74). • Note that in Eq. (2.72) we can take E\ + £2 = £ + •••. Moreover, if /1 is continuous at x\ and fa is continuous at a;2, then in Eq. (2.74) we have d(fiOfa)(x) = {0}. Indeed, in this situation, N(domfa;x~2) = {0} (since x~2 G int(dom/ 2 )); since /1O/2 is continuous at x, d(fiQfa)(x) ^ 0. In particular, it follows that X\ is a minimum point of fa. Furthermore, in this situation, /1O/2 is even Gateaux differentiate. Introducing the convention that 0 • df(x) := N(domf;x) for / G A(X) and x G d o m / , formula (2.75) may be written in the form d(fa0fa)(x)

=

dfa(x1)odfa(x2),

where A o B represents the harmonic sum of the sets A and B, which generalizes the inverse sum of A and B. In order to have more explicit formulas in Corollary 2.8.11 it is necessary to impose some supplementary conditions. Let us give such conditions and the corresponding formulas in three situations for n = 2. Corollary 2.8.15 Let fa, fa G A(X), e G 1+ and

E, >p := m a x { / i , / 2 } . Suppose that one of following conditions is verified: (i) there exists XQ G dorafa n d o m / 2 such that fa is continuous at XQ; (ii) X is a Frechet space, fa, fa are li-convex and 0 G l 6 (dom/i — dom/2); (iii) dim X < 00 and ^dom fa) n *(dom fa) ^ 0.

Convex analysis in locally convex spaces

136

Then, for all x* € X* and x £ dom/i D dom/2 we have: ¥>*(*•) = min{(Ai/i)*(a;I) + (A 2 / 2 )*(^) | (Ai, A2) e A 2 , x\ + x*2 = x*}, dMx)

= \J{dEl(Xifi)(x)+dE2(X2f2){x) e0+ei

I (Ai,A2) G A 2 , e 0 > ei,e 2 > 0,

+ e2 = e, Xifi(x)

dtp(x) = \J{d(X1h)(x)+d(X2f2)(x)

+ X2f2{x) >
I (Ai,A2) G A2>

Xih(x) + x2f2{x) = tp(x)}. Let defined by

-e0},

a

* £ X* and consider the functions / i , / 2 : X —>• K

/i(a;) : = m a x { ( i , a ; ; ) , . . . , ( j ; , < ) } ,

/ 2 (x) := |{a;,a;*)|.

Applying the preceding corollary, we get the following formulas for every xeX: d

h(x)

= { ]C" = 1 A ^i I (Ai, • • •, A„) € A„, ^ = 0 if (x, x*) < /!(a;)} , r {x*} 0/2 (*) = < {Az* I A € [ - 1 , +1]} 1 {-a;*}

2.9

if if if

(x,x*) > 0 , (x, x*) = 0, (x,x*) < 0 .

Convex Optimization with Constraints

Let us come back to the general problem of convex programming as considered in Section 2.5, (P)

min f(x),

x £ C,

where / € A(X), C C X is a convex set and C (~l d o m / ^ 0; the spaces considered in the present section are separated locally convex spaces if not stated explicitly otherwise. Since x is solution of (P) exactly when it minimizes f + ic, x is solution of (P) if and only if 0 € d(f + i
Convex optimization

Proof.

with

137

constraints

In the conditions of our statement we have that

VxeCndom/

: d(f + ic){x) = df(x) + duc{x) = df(x) +

N{C;x),

whence the conclusion is obvious.

•

Very often the set C from problem (P) is introduced as the set of solutions of a system of equalities and/or inequalities. Let Y be ordered by a closed convex cone Q CY and H : X -» Y' be a Q-convex operator; the set C := {x € X \ H(x)
min f{x), H{x)
An element x € X for which H(x)
min f(x),

H(x)


y.

Let us consider the function $ : I

X

F 4 l ,

/(l) *(*,y):=«{ (x, y) := | v wy ' ' oo

i

f

^ ^ ^ otherwise.

(2.76) v '

The problem (Py) becomes now (Py)

min $(x,y),

i £ l

We obtain, without difficulty, that $ is convex. Moreover $*(z*,-y*) = sup{(x,a;*) + (y, -y*) - *(x,y)

\x€X,yeY}

= sup{(x,x*) - (y,y*> - / ( z ) | x € X,y £ Y,#(aO < Q y} = sup{(a;,a;*) -(H(x)+q,y*) = sup{(z,:r*) - (H(x),y*)

- f(x) \ x € dom//, g € Q} - f{x) \ x £

domH}

+ sup{-(g,y*) | 9 € Q}. Hence $*(x*,-y*)

= sup{(x,x*)

if y* e Q+ and $*{x*,-y*)

- f(x) - (H(x),y*)

= oo if y* g Q+.

\x£domH}

138

Convex analysis in locally convex spaces

The function L:XxQ+->R,

L(x,y*):=

{ ^x ^ oo

) + (H(x),y*)

ifxGdomtf, if x $. dom H,

is called the Lagrange function (or Lagrangian) associated to problem (Po)- The above definition of L shows that L(x,y*) — f(x) + (y* o H){x) with the convention that y*(oo) = oo for y* G Q+, convention which is used in the sequel. By what was shown above, we have that = 8up(-L(x,yt)) = -hdL(x,y*). x x xex ^ Therefore, the dual problem of problem (Po) (see Section 2.6) is V y * e Q + : &(0,-y*)

(£>o)

max ( - $*(0,
y*£Y*,

or equivalently, (Do)

max (mfxeX L(x,y*)),

y* £ Q+•

We have the following result. Theorem 2.9.2 Let f G A(X), x G domf and H : X -> (Y',Q) be a Q-convex operator, where Q
3x0edomf

: -H{x0)

€ intQ.

Then the problem (D0) has optimal solutions andv(Po) = v(Do), i.e. there exists y* G Q+ such that inf{/(x) | H(x)
\ x G X}.

Furthermore, the following statements are equivalent: (i) x is a solution of (Po); (ii) H(x)
+ y*o H)(x)

and

(H(x),y*)

= 0;

(hi) there exists y* G Q+ such that (x, y*) is a saddle point for L, i.e. Vx E X, Vy* G Q+ : L(x,y*) < L{x,y*) < L{x,y*)-

(2.77)

Proof. The Slater condition (S) ensures that (XQ, 0) G dom $ and $(x0, •) is continuous at 0, where $ is the function denned by Eq. (2.76). Applying Theorem 2.7.1, there exists y* G Y* such that v(P0) = - $ * ( 0 , - y * ) . If v(Po) = —oo, then $*(0,2/*) = oo for every y* G Y*; in this case we may

Convex optimization with constraints

139

take y* = 0. If f(Po) > —oo, using the expression of $*, we have that y* e Q+. Therefore v(P0) = inf{/(x) | H(x) < Q 0} = - $ * ( 0 , - | T ) = i n f { i ( x , r ) I a; € X } = v(D0). (i) =S- (ii) Of course, x being a solution for (P 0 ), we have that H(x)
< f(x) < f(x) + (H(x),y*)-

(2.78)

Relation (2.78), without the term from its middle, says that x is a minimum point for f + y* o H, and so 0 € d(f + y* o H)(x); taking x = x in relation (2.78) we obtain that (H{x),y*) - 0. (ii) =>• (hi) Since 0 € d(f + y* o H)(x) we have Vx G X : L ( x , r ) = /(x) + (^(x),y*) < f{x) + (H(x),y*)

= LfoiT).

Furthermore, for y* £ Q+ we have that L(x,y*) = f{x) + (H(x),y*)

< f(x) = f{x) + (H{x),y*)

=

L(x,?).

Therefore Eq. (2.77) holds, i.e. (x,y*) is a saddle point of L. (iii) => (i) Taking successively y* = 0 and y* = 2y* on the left-hand side of Eq. (2.77) we obtain that (Hix),y*) = 0. Using again the left-hand side of Eq. (2.77), we obtain that (i?(x), y*) < 0 for every y* e Q+; thus, using the bipolar theorem (Theorem 1.1.9), we have that —i?(x) € Q++ = Q, i.e. x is an admissible solution of (P 0 ). From the right-hand side of relation (2.77) we get Vx e X, H{x)
< f(x) + (H{x),F)

Therefore x is solution of problem (Po)-

<

fix). D

The element y* € Q+ obtained in Theorem 2.9.2 is called a Lagrange multiplier of problem (P). Note that when H is finite-valued (i.e. d o m # = X) and continuous, V x e d o m / : dif +

y*oH)ix)=8fix)+diy*oH)ix);

Convex analysis in locally convex spaces

140

the fact that Q is closed was used only for the implication (iii) =4- (i) of the above theorem, to prove that x is an admissible solution. An important particular case is when there are a finite number of constraints. Let / , g i , . . . ,gn 6 A(X) and consider the problem (Pi)

min f(x),

gi{x) < 0,1 < i < n.

The dual problem of (Pi) is (Di)

max mix€X(f(x)

+ \igi(x)-]

hA n (/ n (i)), Ai > 0 , . . . , A„ > 0.

Then the following result holds. Theorem 2.9.3 dition holds, i.e.

Let f,gi,...,gn

€ A(X). Suppose that the Slater con-

3xo € d o m / , \/i£l,n

: gi(x0) < 0.

Then: (i) the dual problem (Di) has optimal solutions and u(Pi) = i.e. there exist (Lagrange multipliers) A i , . . . , A„ £ M+ such that

v(D{),

mf{f(x)\gi(x)<0,...,gn(x)<0} -ini

{f(x)+J1g1(x)-\

+ Xn9n(x) \ x 6 X) .

(ii) Let x 6 dom / ; x is a solution of problem (Pi) if and only if gi(x) < 0 for every i 6 l , n and there exist A i , . . . , An € K+ such that Xigi(x) = 0 for i £ l , n and 0 € d (/ + Aiffi + . . . +

\XJ-

*ngn)

If the functions g\,..., alent to

gn are continuous at x, the last condition is equiv-

0 € df(x) + A-!0ffi(2c) + . . . +

\ndgn(x).

Proof. Let us consider H : X -> (Mn*,E!J:), H(x) := (gi(x),.. .,gn(x)) for x G n r = i d o m 5 j , i?(x) = oo otherwise; H is R™-convex. The result stated in the theorem is an immediate consequence of the preceding theorem. When gi are continuous at x one has, by Theorem 2.8.7 (iii), that d(f + Aiffi + . . . + \n9n)(x) and d(\igi)(x)

= Jidg^x).

= df(x) + d(Xigi)(x)

+ ... +

d(\ngn)(x), D

Convex optimization with constraints

141

Corollary 2.9.4 Let f,gi,--.,gn '• X —> R be Gateaux differentiable continuous convex functions. Suppose that there exists x$ £ X such that 9i{xo) < 0 for every i 6 l,n. Then x £ X is solution of problem (Pi) if and only if gi(x) < 0 for every i £ l , n and there exist A i , . . . , An £ R+ such that —\7f(x)=\iVgi(x) Corollary 2.9.5 N([g < l\,x)

+ ... + \nVgn(x)

and \igi(x) = 0 for i £ l,n.

•

Let g 6 A(X), 7 £ ] inf g, 00 [ and S £ [5 < 7]. Tften = \J {d(Xg)(x) | A > 0, X(g(x) - 7) = 0}.

(2.79)

Proof. The inclusion "D" in Eq. (2.79) holds without any condition on g and 7. Indeed, let x* £ d(Xg)(x) for some A > 0 with X(g(x)— 7) = 0. Then for x £ [g < 7] we have that (a: — x, x*) < Xg(x) — Xgix) = X(g(x) — 7) < 0, and so x* £ N([g < ^],x). Let x* £ N([g < j];x). Then x is a solution of the problem (Pi)

min {x, —x*), h(x) := g(x) — 7 < 0,

whence, by Theorem 2.9.3, there exists A > 0 such that Xh(x) = X(g(x) — 7) = 0 and 0 G d(-x* + Xh){x), i.e. x* £ d{Xh){x) = d(Xg)(x). Therefore the inclusion " c " of Eq. (2.79) holds, too. • Note that d(Xg)(x) = Xdg(x) if A > 0 and d(0g)(x) = didomg(x) = N(domg;x). In the case of normed vector spaces, taking A = X in Proposition 3.10.16 from Section 3.10, we have another sufficient condition for the validity of formula (2.79). Note that we can obtain the characterization of optimal solutions in Theorem 2.9.3, when the functions gi are continuous, using Corollary 2.9.5 and the formula for the subdifferential of a sum. Indeed, x £ dom / is a solution of (Pi) if and only if x is minimum point of the function / + icx + htc„, where Cj := {x | gi(x) < 0}. By hypothesis d o m / f l f)" = 1 int d ^ 0, and so, for every x £ d o m / (~l H i l i ^ i w e have d(f + tCl+...

+ icn) (x) = df{x) + N(Cf,x)

+ ... +

N(Cn;x).

Using formula (2.79) we obtain the desired characterization. Theorem 2.9.2 can be extended further to the case when there are also linear constraints. Let us consider the problem (P 2 )

min f(x),

H{x)
Convex analysis in locally convex spaces

142

where T G £ ( X , Z). Consider the Lagrange function associated to problem

L2 : X x (Q+ x Z*) - • 1 ,

L 2 ( a M , V ) := /(a:) + (H(x),y*)

+

(Tx,z*).

The dual problem associated to (P 2 ) is (£>2)

max (mixeXL2(x,y*,z*)),

y* G Q+, z* G Z*.

Then the following result holds. Theorem 2.9.6 Let f G A{X), H : X -> (Y,Q) be a Q-convex operator, where Q C Y is a closed convex cone, T G L(X,Z) be a relatively open operator and x G d o m / . Suppose that there exists XQ € d o m / such that f,H are continuous at xo, —H{XQ) G int<5 and TXQ = 0. Then the problem (D2) has optimal solutions and V{PQ,) = v{D2), i.e. there exists y* G Q+, z* G Z* such that inf{/(x) I H(x)
\ x G X).

Furthermore, the following statements are equivalent: (i) x is solution of problem (P2); (ii) H(x)
and (H(x),y*)

= 0;

(iii) there exists (y*,z*) G Q+ x Z* such that (x,(y*,z*)) point of L2, i.e. L2(x,y*,z*)

< L2{x,y*X)

<

is a saddle

L2{x,y*,T)

for all x G X and all y* G Q+, z* G Z*. Proof.

Let us consider the perturbation function

*:Ix(7xZ)->I, v

;

'

f(x) $(1,y,z) := , y y ^ ' ' ' 00

if H(x)
We intend to prove that condition (i) of Theorem 2.7.1 is verified. Note first that P r y x Z ( d o m $ ) = {{H{x) + q,Tx) | x G d o m / , q £ Q} C Y x ImT.

A minimax

theorem

143

Taking into account that / is continuous at xo G d o m / , there exists Uo G N x ( i o ) such that Vxe%

: f{x)<M:=f(x0)

+ l.

Since —H{XQ) G intQ, there exists Vo G Ny such that —H(XQ) +VQCQ. There exists V € Ny such that V + V CV0. Since if is continuous at x0, there exists {/ € Nx(zo)) [/ C t/o, such that H(x) € H(x0) — V for every x € U. Since T is relatively open, there exists W G NimT such that W C T(LT). Of course, V x W G N y x i m r ( 0 , 0 ) ; let (j/,z) G V x W. There exists x £U such that Tx = z. Since x G f/, we have that i ? ( i ) = H(xo) — y' for some j / ' G V. Then y - H(x) =y + y'-

H(x0) G - # ( x 0 ) + V + V C - # ( x 0 ) + V0 C Q;

hence -ff(x) < Q y. Since x £ U C Uo, we have that $(x,2/,z) < M, i.e. condition (i) of Theorem 2.7.1 is verified. Therefore there exists (y*,z*) G Y* x Z* such that v(P2) = -4>*(0, -y*, -z*). But $*(0, -y*, -z") = s u p ( w ) 6 X x y x Z ( ( i / , -y*) + (z, -z*) - $ ( x , y , z)) = sup{-(y,y*)

- (z,z*) - f(x) | H ( i )
= s u p { - ( # (x) + q, y*) - (Tx, z*) - f(x) = - inf{f(x) + (H(x),y*)

\xeX,qeQ}

+ (Tx, z") \ x G X } + sup{-(q,j/*) | q G Q}.

Thus **cn _„* —?*\ - J - i n f i e x ^ C i . y * , ^ * ) * l U ' y ' z ) ~ \ oo

if if

2/* e Q + , j / * £ Q+.

Therefore y* G Q + if u(P 2 ) G K and we can take j7* = 0 if v(P2) = - o o . The proof of the second part is completely similar to that of Theorem 2.9.2. We note only that, / and H being continuous at xo, we have d(f + y* o H)(x) = df(x) + d(y* o H)(x) for every x G d o m / . •

2.10

A Minimax Theorem

In the preceding section we have obtained some results which are stated using saddle points. In the sequel we establish a quite general result on the

144

Convex analysis in locally convex spaces

existence of saddle points as well as a minimax theorem. Let A and B be two nonempty sets and / : A x B -> JR. It is obvious that sup inf f(x,y) xeAvtB

< inf sup f(x,y). y£BxeA

(2.80)

The results which ensure equality in the preceding inequality are called "minimax theorems," the common value being called saddle value. Note that if / has a saddle point, i.e. there exists (x,y) € A x B such that VxeA,Vy&B

: f(x,y)

< f(x,y)

< f(x,y),

(2.81)

then max inf f(x,y) x£A yeB

= minsup f{x,y), yeB

(2.82)

xeA

where max (min) means, as usual, an attained supremum (infimum). Indeed, let (x,y) € Ax B satisfy Eq. (2.81). It follows that ini: sup f(x,y) v£Bx€A

< sup f(x,y) xeA

< f(x,y)

< infB :(x,y) < sup infB y£ xeAv&

f{x,y).

Using Eq. (2.80) we obtain that all the terms are equal in the preceding relation, and so (2.82) holds (the maximum being attained at x and the minimum at y). Conversely, if Eq. (2.82) holds then / has saddle points. Indeed, let x G A and y e B be such that inf (x,y) = sup ini' f(x,y) y£B

xeAVZB

= irtf swp f(x,y) y€Bx€A

= sup

f(x,y).

x£A

Since inf y 6 s(x,y) < f(x,y) < supx€Af(x,y), from the above relation it follows that all the terms are equal in these inequalities; this shows that (x, y) is a saddle point of / . So we have obtained the following result. Theorem 2.10.1 Let A and B be two nonempty sets and f : A x B ->• R. Then f has saddle points if and only if condition (2.82) holds. • The following result is an enough general minimax theorem (frequently utilized) which gives also a sufficient condition for the existence of saddle points. Theorem 2.10.2 Let X be a locally convex space, Y be a linear space, Ac X be a nonempty convex compact set and B CY be a nonempty convex set. Let also f : A x B -)• M. be a function with the property that f(-,y)

A minimax

theorem

145

is concave and upper semicontinuous for every y G B and f(x, •) is convex for every x E A. Then max inf f(x,y)

— inf ma,xf(x,y).

X&A y£B

y£B

(2.83)

xEA

If moreover Y is a locally convex space, B is compact and f(x, •) is lower semicontinuous for every x € A, then maxmmf(x,y)

= min max f(x,y);

xeA y£B

yeB

(2.84)

x£A

in particular f has saddle points. Proof. Let a e K be such that a > max x£j 4 inf ye B f(x,y). For every x S A there exists yx 6 B such that f(x,yx) < a. Since f(-,yx) is upper semicontinuous at x, there exists an open neighborhood Vx of x such that f(u,yx) < a for every u G Vx. Since A is compact and A C UxeA ^ > there exist x\,..., xp € A such that A C U?=i Vxt • Let 2/j := yXi • Consider the sets d

:=co{{f(x,yi),...,f(x,yp))\xeA}cW,

C2 := {(ui,...,up)

\v,i>a\/i€

l,p} .

It is obvious that C\ and Ci are nonempty convex sets, and C2 has nonempty interior. Moreover C\ fl Ci = 0. In the contrary case there exist q 6 N, ( A i , . . . , A?) 6 Aq and Xi,..., xq € A such that V i e l . p : a < Zf := 2^,

>^jf{xj,yi).

Since f(-,yi) is concave, taking xo = 13?=i ^ j 2 ^ ' Vi G l , p : a < Zj <

we nave

that i o £ i and

f(x0,yi).

There exists io £ l , p such that £0 £ VXi . Therefore f{xo,yi0) < a, a contradiction. Applying Theorem 1.1.3 there exists (/zi,..., fj,p) € W \ {0} such that . ,mf(x,yi) t=l

< V

UiUi.

*—'i=l

Letting u; -> 00 we have that /ij > 0 for every i, and so we can suppose that (/xi,..., Up) € A p . Taking u = ( a , . . . , a) we obtain that ^xeA

: > .

Uif{x,yi)

< a.

Convex analysis in locally convex spaces

146

Let 2/0 := Y%=i ViVi € B. Since f(x, •) is convex, we obtain t h a t f(x, y0) < a for every x € A. Therefore a > i n f j / e B m a x x g ^ / ( a ; , 2 / ) , and so (2.83) holds. In Eq. (2.83) the suprema with respect t o x are attained because the functions f{-,y), y £ B, and i n f y e s f(-,y) are upper semicontinuous and A is compact. W h e n B is compact and the functions f(x,-) are lsc for all x € A we obtain t h a t the infima with respect to y € B are attained in Eq. (2.83), i.e. Eq. (2.84) holds. • Note t h a t the convexity assumptions in the preceding theorem can be weakened. More precisely, we can suppose t h a t A is a nonempty compact subset of a topological space, B is nonempty, and / satisfies t h e following two conditions (similar to concavity and convexity, respectively):

Vxi,...,xp

e A, V ( A i , . . . , A p ) e A p , 3x0

&A,\fyeB

:

f(xo,y) > y \

,Kf(zi,y),

• ' — ' 1 = 1

\/yi,...,yq

GB,

V ( / / ! , . . . , / * , ) € A , , 3y0 eB,Vx£A

: v—yQ

f(x,yo) 2.11

<

2^,.=iHjf{x,yj).

Exercises

Exercise 2.1 Let X be a linear space, / € A.(X) and x,u £ X. Prove that the mapping %p :]0, oo[—• R defined by ip(t) :— t • f(x + t~lu) is convex. E x e r c i s e 2.2 (a) Let I C R be an interval and / : / — > • R. Suppose that / is locally nondecreasing, i.e. for every a € I there exists e > 0 such that the restriction of / to IC\[a — s, a + e] is nondecreasing. Prove that / is nondecreasing. (b) Let g : [a, b] —• R (a, 6 € R, a < b) be a continuous function. 1) Assume that g'+{x) g R exists for every x € [a,b[; show that there exists x' £ [a,b[ such that g(b) — g(a) < g'+(x')(b — a). 2) Assume that g'-(x) € R exists for every x S ]a, 6]; show that there exists x" € ]a, b] such that g(b) — g(a) > gL (x")(b — a). (c) Let X be a locally convex space and A C X be an open convex set. If / : A —> R is locally convex, i.e. for every a G A there exists a neighborhood V of a such that f\vnA is convex, prove that / is convex. Exercise 2.3

Let / : R n -*• R be a quasi-convex function and i ^ f .

For

Exercises x eRn

147

consider the function fx : R n ->• R, fx(t) = f(x + tx). Prove that / is lsc at i t t V i e l "

: fx

is lsc at 0,

/ is use at J o V i e R "

: fx, is use at 0.

If R n is replaced by an infinite dimensional normed space then the above properties do not hold, generally. Exercise 2.4 Let X be a linear space, / : X —> R be a convex function and A > i n f i e x fix)- Prove that {x G X | f(x) < \y

= {x G X | f{x) < A}.

Moreover, if X is a topological vector space and / is continuous, then int{x € X | f(x) < A} = {x G X | / ( z ) < A}. Exercise 2.5 Let X be a topological vector space and / : X —> R a convex function. Prove that: ( a ) [ / < * ] = c l [ / < *] f° r every t G ] inf / , oo[ if and only if / is lsc; (b) [ / < * ] = i n t [ / < *] for every t G ] inf / , cx)[ if and only if / is continuous on d o m / ; ( c ) [/ = *] = b d [ / < t] for every t G ] inf / , oo[ if and only if / is continuous (on X ) . Exercise 2.6 Let / : R" —)• R be a proper convex function for which all the partial derivatives df/dxi exists at a G int(dom/). Prove that / is Gateaux differentiable at a (even Frechet differentiate because / is Lipschitz on a neighborhood of a). Exercise |1 + t\p + is strictly and (f2 is

2.7 Let p G [1, oo[ and R be defined by tpp(t) := |1 - t\p 2pt. Prove that ipp is strictly convex and increasing for p G]l, 2[, ipp concave and decreasing for p G ]2, oo[, ip\ is convex and nondecreasing constant.

E x e r c i s e 2.8

Take /3 G [1, <x[ and consider the function

/:R2^I,

2

f(x,y):={ (^tan|)^V^T^ [ oo

for x, y > 0, otherwise,

where, by convention, arctan | := ^ for a; > 0. Prove that / is a lsc convex function, but not strictly convex. Exercise 2.9

Consider the function

/: c[o, i] -> R,

fix) := /o1 y r r w o F ^ -

Convex analysis in locally convex spaces

148

Prove that / is convex and Frechet differentiable of order 2 on C[0,1] with

V/(x)(«) = f1 -£L= dt, Jo V l + x

V2f(X)(u, V) = ^

2

,7/^3-2 dt 2

Jo ( l + 2 : ) v l + z 2

for all u,v £ C[0,1]. Moreover, V / is continuous on C[0,1]. E x e r c i s e 2.10

Consider the function /:L1(0,1)->R,

/ ( i ) : = / 0 V l + (i(t))!*,

where L 1 (0,1) is the Banach space of (classes of) Lebesgue integrable functions on the interval [0,1]. Prove that / is a Gateaux differentiable convex function with xu Vx.uSL^O.l) : V/(x)(u) = / :dt, Jo

%/rn

but / is nowhere Frechet differentiable. Exercise 2.11

Using properties of convex functions, prove that

V a , 6 e R + , V p , g e ] l , o o [ , £ + ^ = 1 : ab < ±a p + \b\ the equality being valid if and only if ap = bq, and V n € N, V x i , . . . , x „ 6 R + , V a i , . . . , a „ 6]0,1[, a i + . . . + a „ = 1 : a m + . . . + a n x „ > X™1 • • -xZn, the equality being valid if and only if x\ = • • • = xn. In particular, VneN, Vsi,...,x„ €R+

: £(xi + . . . + xn) >

yX\---xn.

E x e r c i s e 2.12 Let / : X —>• R be a proper function. Prove that / is convex if and only if / + x* is quasi-convex for every x* G X*. Exercise 2.13 Let <x\,... ,an > 0 (n > 1) be such that ai-\ \-an < 1. Prove that the function / : P " —> R, / ( x ) := x"1 • x%2 • • • x"n, is concave; moreover, if a\ + • • • + a„ < 1, then / is strictly concave. E x e r c i s e 2.14 Consider / : R n ->• R, f(xi,...,x„) Prove that / is convex. Exercise 2.15

:= In ( X £ = 1

expxk).

Consider p € R \ {0} and the function xP"->R,

f{t,x):=

t"

nr=i*i'

Prove that / is strictly convex for p € ] — o o , 0 [ U ] n + l , oo[ and convex if p = n + 1.

149

Exercises Let c G R n and A : = { i £ P " \(x\c)> g : A -> R,

0}. Prove that the function

p(x)

(x\c)"

niu^

is strictly convex for p > n + 1 (it is possible to prove that g is strictly convex for p > n). The function g has been used by Karmarkar for establishing his interior point algorithm. Finally prove that the function

h : P n ->• R,

h(x) := (T[n xA

is strictly convex. Exercise 2.16 Let tp : R+ —> R+ be a continuous convex function such that ^i(t) = 0 •«• t = 0. Show that

Jo

i>'- (V>- x «))

=

7o

V-H^-H*))

for every a > 0. Exercise 2.17 function

Let X, Y be normed spaces and T G L(X,Y). f:Y^R,

f(y):=M{\\x\\\Tx

Prove that the

= y},

is a sublinear functional. Moreover, if T is an open operator, then dom f = X and / is continuous. Exercise 2.18 Let / : Rk —> R be a strongly coercive proper lsc function. Prove that co(epi/) is a closed set. E x e r c i s e 2.19 Let X be a locally convex space, / G T(X), g : X —> R be a proper function and a, /3 > 0. Prove that co (co(/ + ag) + jig) = co ( / + ( a 4- /3)g) and ( ( / + ag)** + fig)" = (f + (a + fig)* . Exercise 2.20 Let X be a separated locally convex space and A,B,CcXbe nonempty sets. If C is bounded and A + C C B + C prove that A C coB. Exercise 2.21 Let (X, \\-\\) be a normed space, / G T(X) and L > 0. Prove the equivalence of the following statements: (i) d o m / = X a n d V x , t / G X (ii) 3 a GR, V i € X (iii)

VMGX

: \f(x) - f(y)\ < L \\x - y||;

: / ( x ) < L ||x|| + a;

: /oo(u)
Exercise 2.22 Let X be a locally convex space and / € T(X). foo(u) = sup{/(z + u) — f(x) I a; G d o m / } for every u 6 l .

Prove that

150

Convex analysis in locally convex spaces

E x e r c i s e 2.23 Let X, Y be separated locally convex spaces, / € r ( X ) , F € F(X x Y), x € dom / and A £ R, e £ R+ be such that [/ < A] and d£f(x) are nonempty. Prove that: [/ < A]oo = [foo < 0], ( a e / ( x ) ) 0 0 = N{domf;x), (/*)«, = Sdom/, foe = Sdom/* and {v* £ y* | (F*)oo(0,t;*) < 0} = ( P r y ( d o m F ) ) . In particular, if {v* £ Y* | (F*) o o (0,u*) < 0} is a linear subspace then {0} and Pry (dom F) are united. E x e r c i s e 2.24 Let X be a separated locally convex space and / £ I \ X ) . Consider K := {u £ X | /«,(w) < 0} and X 0 := K D (-.K"). Prove that: (i) if is a closed convex cone, Xo is a closed linear subspace and f(x + u) = f(x) for all a: 6 X and u 6 Xo. (ii)

lin(dom/*)=X6L.

(iii) The function / : X / X 0 -»• I , f(x) := f(x), is well defined, fe r ( X / X 0 ) , /OO(M) = /oo(«) for every u € X, f* = f*\x± and 5 £ / ( £ ) = 9 E /(a;)| X j. for all x € X and £ 6 R+, where x represents the class of x £ X. E x e r c i s e 2.25 Let X be a separated locally convex space and / £ T(X). Assume that there exists u £ X such that /<X>(M) < 0 < fco(-u). Prove that for every e > 0 there exists fe £ T(X) such that f(x) < fc(x) < f(x) + s for every x e X and argmin/ e = 0. E x e r c i s e 2.26 Let X, Y be separated locally convex spaces and F 6 A(X x Y). Assume that F satisfies one of the conditions (ii)-(viii) of Theorem 2.7.1. Prove that the marginal function g : X* —¥ R, g(x") := inf y * e y* F*(x*,y*) is convex, iu*-lsc, the infimum is attained for every x* £ X* and goo(u") = mmv*ey(F*)oo(u* ,v*). Moreover, {v* £ Y* \ (F*)oo(0, «*) < 0} is a linear subspace. E x e r c i s e 2.27 (Toland-Singer duality formula) Let X be a separated locally convex space, / : X —>• R be a proper function and g £ T(X). Then inf (/(*) - ff(i)) = .inf. (g\x')

- f*{x'))

•

E x e r c i s e 2.28 Let X be a separated locally convex space, / £ T(X) be such that 0 € d o m / a n d p £ P. Consider the following assertions: (i) the mapping P 3 t i - > t~pf(tx) (ii) f'(x, —x) +pf(x) (iii) pf(x) (iv)

is nondecreasing for every x £ d o m / ;

< 0 for every x £ d o m / ;

< f'(x,x)

for every x € d o m / ;

{x,x*) >pf(x)

for all (x,x*) € g r d / ;

(v) for every x £ int(dom / ) there exists x* 6 df(x) such that (x, s*) > pf(x). Prove that (i) <=> (ii) <=> (iii) =$- (iv). Moreover, if / is continuous at 0 then all five assertions above are equivalent.

Exercises

151

E x e r c i s e 2.29 Let / : X —> IR be a function such that c o / is proper. Assume that for some x € d o m c o / we have that x = ^2i=i ^»^« a n ( ^ co/(ic) = ^ * = 1 ~\if{x~i) with Xi G d o m / , Ai > 0 for i € ITfc and £)* = 1 ^> = 1- Prove that 8c5/(3f) = n t i df(xi). Exercise 2.30 Let X be a separated locally convex space and / € A(X). Assume that 0 G d o m / and / | x 0 is continuous at 0, where Xo •— aff(dom/). Prove that f = sup{(x,x*) |x* G<9/(0)W < { =

f'(0,x) f'(Q,x) = oo / ' ( 0 , x ) = oo

if if if

xGXo, xGXo\_Xo, xeX\X0,

the supremum being attained for x G Xo. Moreover, Xo is closed if and only if V x G X : / ' ( 0 , x ) = s u p { ( x , x * ) | x* 6 3/(0)} . E x e r c i s e 2.31 Let X be a separated locally convex space and / G A(X) be continuous on i n t ( d o m / ) , supposed to be nonempty. Prove that for all x,y € int(dom/) there exist z £]x,y[and z* G df{z) such that f(y)—f{x) = (y — x, z*). E x e r c i s e 2.32

Let X be a separated locally convex space and / G T(X).

(i) Consider the conditions: a) df is single valued on dom.3/, b) (df)~1(xl)!~) (df)~1(x2) = 0 for all distinct elements Xi,X2 G X*, c) /* is strictly convex on every segment [xi,X2] C Imdf, d) dom df = int(dom/). Prove that a) <=> b) •<=>• c); moreover, if int(dom/) ^ 0 and / is continuous on int(dom/) then a) => d). (ii) If / is continuous on int(dom/) and ( 9 / ) _ 1 is single-valued on show that / is strictly convex on int(dom/).

Imdf,

E x e r c i s e 2.33 Let X, Y be separated locally convex spaces and / G A(X x Y). Prove that if / is continuous at (xo,j/o) £ d o m / , then Prx* (df{x0, j/o)) = df(-,yo)(x0) Exercise 2.34 Consider

and

P r y ( d / ( x 0 , y 0 )) = d / ( x 0 , -){yo)-

Let X be a separated locally convex space and / o , / i 6 A(X).

v := inf{/ 0 (x) | / i ( x ) < 0},

v* := sup inf (/o(x) + A/i(x)), x>o xex

with the convention 0 • oo = oo. Suppose that » £ R . Prove that v* =v&[rfe>Q

: inf{/i(x) | / 0 (x) < y - e} > 0].

E x e r c i s e 2.35 Let {X, \\-\\) be a normed linear space and / : X —> R be a proper convex function. Prove that there exists x* G X* with x* < / if and only if there exists M > 0 such that f(x) + M \\x\\ > 0 for all x G X.

Convex analysis in locally convex spaces

152

Exercise 2.36 Let (X, ||-||) be a normed linear space and / i , / 2 : X -> R be proper convex functions. Prove that there exist x* 6 X* and a € R such that —/i < x" + a < J2 if and only if there exists M > 0 such that fi(xi) + f2(%2) + M\\xi - x21| > 0 for all on, x2 € X. E x e r c i s e 2.37 Let X be a linear space, (Y, ||-||) be a normed linear space, T : X -* Y be a linear operator, yo 6 Y and / : I -> R be a proper convex function. Prove that f(x) + \\Tx + yo\\2 > 0 for all x € X if and only if there exists j / * € y * such that fix) -2(Tx + y0,y*) - ||y*|| 2 > 0 for all x e X. E x e r c i s e 2.38 Let (X, ||||) be a normed vector space, C,D C X be closed convex cones and x G X, x* € X*. Prove that d(x,C)

: = i n f { | | x - x | | | x € C} = m a x { - ( x , x * ) | x* 6 Ux* n C + } ,

d ( x * , C + ) = m i n { | | x * - x * | | | a;* 6 C + } = s u p { - ( x , x * ) | x G C/x n C} , a,ndsupx€Uxncd(x,D)

= supx,eUx,nD+

d(x* ,C+).

E x e r c i s e 2.39 Let / € A(X) and x € d o m / be such that f(x) Consider the sets d
>

inf/.

:= {z* € X* | ( i - x,x*) < f(x) - /(a?) Vx e [/ < /(*)]},

d ^ / ( x ) := {x* a ' l f i - f , x*) < f{x) - / ( J ) Vx € [/ < /(*)]}. Prove that dKf{x) = d-f(x) differential of / at x.

= [1, oo[-d/(x). The set a < / ( x ) is Plastria's sub-

E x e r c i s e 2.40 Let X be a normed space, (an)n>i C X and (A n )„>i C [0,oo[ be such that ^ ^ L j A n = 1. Consider the function

/:X->R,

f{x) = Y°°

A„||x-a„|| 2 .

*•—'n=l

1) Prove that the following statements are equivalent: (a) S^Li-^i|l a "ll 2 < oo (i.e. 0 € d o m / ) , (b) d o m / ^ 0, (c) d o m / = X. 2) Prove that / is finite, convex and continuous when 5Z^Li^ill a "l| 2 < °°3) Suppose that ]C!!!Li^n|lan||2 < °°- Prove that for every x £ X one has 8f{x) = { 2 V 0 0

A n ||x - On|| xmn I x*n e a|| • IKi - a„) Vn € N ) .

E x e r c i s e 2.41 Let X be a normed space and / 6 T(X). following statements are equivalent:

Prove that the

(a) lim||J.||_too f{x) = 00; (b)

[/ < A] is a bounded set for every A > inf^ex / ( x ) ;

(c) there exists Ao > inf^gx / ( x ) such that [/ < Ao] is bounded;

Exercises

153

(d) there exists a, 0 £ R, a > 0, such that f(x) > a||x|| + /? for every x G X; (e) liminf| )x ||_ foo / ( x ) / | | x | | > 0; (f)

0£int(dom/*).

Moreover, if dim X < oo then the conditions above are equivalent to ( c ')

[/ < iaf /] is nonempty and bounded.

Furthermore, if p, q £ ]l, oo[ are such that 1/p + 1/q = 1, the following statements are equivalent: (g)

liminf|NHoo/(x)/||x||p>0;

(h) l i m s u p | | : c , | H o o r(x")/\\x*\\q

< oo;

(i) there exists a, /3 £ R, a > 0, such that f(x) > a||x|| p + P for every i 6 X ; (j) there exists a,/3 £ R, a > 0, such that /*(x*) < cx||a;*||9 + /? for every a;* e X * . E x e r c i s e 2.42 Let f,fn : R m ->• R (n £ N) be convex functions such that (fn{x)) -¥ f{x) for every x £ R m . Assume that / is coercive. Prove that there exist a,P £ R with a > 0, no £ N such that / n ( x ) > a \\x\\ + /3 for all x G R m and n > no. Exercise 2.43 Let (X, ||||) be a n.v.s. and C C X be a nonempty closed convex set. Consider the following assertions: (i) there exists XQ G X* such that (x,x*,) > \\x\\ for every x G C; (ii) there exist xo £ -X", X*, £ X* such that (x — xo, x*>) > ||x — xo|| for every x eC; (iii) int(dom sc) 7^ 0; (iv) a) there exists x*, G X* such that {u, x*>) > 0 for every u £ Coo \ {0} and b) for every sequence (x n ) C C with (||x n ||) —• oo and ( | | x „ | | _ 1 xn) -^ u one has Prove that (i) => (ii) =S> (iii) => (iv). If 0 £ C then (ii) =>• (i). Moreover, if X is a reflexive Banach space then (iv) =>• (ii). Exercise 2.44 Let X be a normed space and / £ A(X) be lower bounded. Consider A > i n f l € x / ( x ) =: inf / and p > 0. We envisage the conditions: (a)

[f<X}CpU;

(b) V x £ X (c) V x ' G

A

: f(x) > inf / + "2'°f/t/x»

A

~lnf 2p

f

• max{0, l|a?l| - p};

: / * ( * ' ) < / * ( 0 ) + p||x'||.

Prove that (a) =>• (b) & (c). Exercise 2.45 equivalences:

Let X be a normed space and / G T(X). Prove the following

Convex analysis in locally convex spaces

154

(a) liminf||.1.||_+00 / ( x ) / | | x | | > 0 - » 0 £ int(dom/*); in this case f(x) liminf 4f-if = sup{/i > 0 | /* is upper bounded on (J.U*}. ||x||-KX>

||X||

(b) liminf || B | H o o f(x)/\\x\\

= 0 « 0 6 Bd(dom/*).

(c) liminfHJII-KX, f(x)/\\x\\

< 0 O 0 ^ cl(dom/*); in this case

f(x)

liminf ~-r{- = - d d o m / . (0). llxH-KX)

E x e r c i s e 2.46

||Z||

Let p 6 ]1, oo[ and f:£p^R,

/ ( ( * „ ) „ > i ) := Y°°

n\xn\n.

Prove that / is finite, convex, continuous and lim||„||_+0O f*(y)/\\y\\

— 1.

E x e r c i s e 2.47 Let X, Y be Hilbert spaces, C C X be a nonempty closed convex set and A G L(X,Y) be a surjective operator. Consider the problem min i||Ac|| 2 ,

(P)

xeC.

A solution x of problem (P) is called a spline function in C associated to A. (a) Prove that (P) has optimal solutions if C + ker ^4 is closed. (b) Prove that x G C is an optimal solution of (P) if and only if y := A* (Ax) satisfies the relation (x\y) = min{(x \y) \ x € C}. E x e r c i s e 2.48 Let X,Y be Hilbert spaces, C := {x 6 X | Vi, 1 < i < k : (x\a,i) < fii}, where ai,...,ajb G -X', Pi,...,Pk G R and A G £ ( X , F ) be a surjective operator. Consider the problem (P)

min

5 | | A E | | 2 , x G C.

Prove that (P) has optimal solutions. Suppose, moreover, that there exists x € X such that (x \ a{) < /3i for every i, 1 < i < k. Prove that l i s a solution of (P) if and only if there exists (\i)i
V Xkdk a n d V i , 1 < i < A: : A;((x| a») — /?;) = 0.

Exercise 2.49 Let X be a Hilbert space and 01,02,03 be three non colinear elements of X (i.e. they are not situated on the same straight-line). Prove that there exists a unique element x G X such that V i £ l

: | | x - O i | | + | | x - a 2 | | + | | i - a 3 | | < ||x - ai|| + ||x - o 2 || + ||x - a 3 ||.

Moreover, prove that x G co{oi,a2,03}. The point x is called the Toricelli point of the triangle of vertices a\, 02,03.

Bibliographical notes

155

E x e r c i s e 2.50 Determine the optimal solutions (when they exist) and the value of the problem (Pf)

min £ (tx{t) + fiy/1 + (u(t)) 2 ) dt, x e Xi,

for every fj, € R + and every i g {1, 2, 3,4}, where Xi:=C[0,l], X 3 := | x e C [ 0 , l ] | / o 1 x ( t ) d t = 0 J , E x e r c i s e 2.51 (P)

X a : = 2/(0,1), X» := { x G 1/(0,1) | /^ x(t)dt = o } .

Consider the (convex) optimization problem

max J,,1 x(t) dt, xeX,

x(0) = x(l) = 0, J,,1 y/l +{x'(t))2dt

< L,

where L > 0 and X = C^O, 1] := {x : [0,1] —• R | x derivable, x continuous on [0,1]} or X = AC[0,1] := {x : [0,1] -¥ R | x absolutely continuous}. Determine the optimal solutions of problem (P), when they exist, and its value (using, eventually, the dual problem).

2.12

Bibliographical N o t e s

Many results of this chapter are well-known and can be found in several books treating convex analysis: [Rockafellar (1970); Hiriart-Urruty and Lemarechal (1993)] (for finite dimensional spaces), [Ekeland and Temam (1974); Ioffe and Tikhomirov (1974); Barbu and Precupanu (1986); Castaing and Valadier (1977); Phelps (1989); Aze (1997)]. In the sequel we point only those results that are not contained in these books but the last one. Sections 2.1-2.5: The assertion (vii) of Theorem 2.1.5 was established in [Zalinescu (1983b)]. The characterization of convex functions using upper Dini directional derivatives in Theorem 2.1.16 as well as Lemma 2.1.15 are established by Luc and Swaminathan (1993). Theorem 2.2.2 was established by Crouzeix (1980) in finite dimensional spaces and Theorem 2.2.11 by Zalinescu (1978). The notion of quasi-continuous function was introduced by Joly and Laurent (1971); Proposition 2.2.15 and Corollary 2.2.16 are established by Moussaoui and Voile (1997). The notions of cs-convex, cs-closed and cs-complete functions are introduced by Simons (1990), while that of lcs-closed function by Amara and CiligotTravain (1999); all the results concerning lcs-closed functions are due to Amara and Ciligot-Travain (1999). Theorem 2.2.22 (for I = N) and Proposition 2.2.24 can be found in [Kosmol and Muller-Wichards (2001)]; when d i m X < oo Theorem 2.2.22, Corollary 2.2.23 and Proposition 2.2.24 are stated in [Marti (1977)] under the weaker hypothesis that the pointwise boundedness or pointwise convergence holds on a dense subset of C. Proposition 2.2.17 is stated in [Zalinescu (1992b)]. Proposition 2.4.3 is established by Rockafellar (1966), Theorem 2.4.11 is proved in the book [Rockafellar (1970)] in R n and stated by Hiriart-Urruty

156

Convex analysis in locally convex spaces

(1982) in the general case; one can find another proof in [Aze (1997)]. The local boundedness of df in Theorem 2.4.13 and Corollary 2.4.10 can be found in many books on convex analysis. The assertions (iv) and (vii) of Theorem 2.4.14 are established in [Zalinescu (1980)]; for (i) see also [Anger and Lembcke (1974)]. Theorem 2.4.18 is from the book [Ioffe and Tikhomirov (1974)]. Theorem 2.5.2 was proved by Polyak (1966) under the more stringent conditions that / is a quasi-convex function which is bounded and Lipschitz on bounded sets and attains its infimum on every closed convex subset of X. Theorem 2.5.5 and Lemma 2.5.3 are established by Borwein and Kortezov (2001), while Lemma 2.5.4 and other results on non-attaining convex functionals are established by Adly et al. (2001a). Sections 2.6-2.9: The systematic use of perturbation functions for calculating conjugates and subdifferentials was done for the first time by Rockafellar (1974). The author of this book continued this approach in [Zalinescu (1983a); Zalinescu (1987); Zalinescu (1989); Zalinescu (1992a); Zalinescu (1992b); Zalinescu (1999)]; this permitted, for example, to give simpler proofs to several results stated in [Kutateladze (1977); Kutateladze (1979a); Kutateladze (1979b)]. Theorem 2.6.2(i) was established by Moussaoui and Seeger (1994), but Theorem 2.6.2(h) and Theorem 2.6.3 are new. Corollaries 2.6.4-2.6.7 can be found in [Hiriart-Urruty and Phelps (1993)] and [Moussaoui and Seeger (1994)]; for other results in this direction see the survey paper [Hiriart-Urruty et al. (1995)]. The sufficient conditions for the fundamental duality formula and for the validity of the formulas for conjugates and e-subdifferentials are, mainly, those from author's survey paper [Zalinescu (1999)], where one can find detailed historical notes; here we mention only the first use (to our knowledge) of them; actually, all these sufficient conditions are mentioned in that paper, excepting those which use li-convex or lcs-closed functions. So, conditions (iii) and (viii) of Theorem 2.7.1 and the corresponding ones in Theorems 2.8.1, 2.8.3, 2.8.7 and 2.8.10 are the classical ones and can be found in all the mentioned books which treat them. Condition (i) of Theorem 2.7.1 was used by Rockafellar (1974) when Yo = Y and by Zalinescu (1998) in the present form (see also [Combari et al. (1999)]), (ii) and (vi) were introduced in [Zalinescu (1983a)] for Yo = Y (and R replaced by a separated lcs ordered by a normal cone) and in [Zalinescu (1999)] in the present form, (iv) was introduced in [Zalinescu (1992b)] with (Hwi) replaced by (Hx), (v) is stated by Amara and Ciligot-Travain (1999) for X, Y Frechet spaces, $ a lcs-closed function and ,b replaced by lc, (vii) was introduced by Rockafellar (1974) for X, Y Banach spaces (X even reflexive) and Yo = Y, and by Zalinescu (1987) in the present form, while (ix) was used by Joly and Laurent (1971) for $ lsc and by Moussaoui and Voile (1997) for arbitrary 3>; note that condition (b) of [Cominetti (1994), Th. 2.2] implies condition (2.54). Theorem 2.7.4(iv) was established in [Zalinescu (1992b)]; the other conditions were introduced in [Zalinescu (1999)] (but in (v) it was assumed that F is lsc and C is closed). Condition (iii) of Theorem 2.8.1 was introduced in [Ekeland and Temam (1974)], (vii) was introduced in [Zalinescu (1984)] (for Y0 = Y), (iv) was intro-

Bibliographical notes

157

duced in [Zalinescu (1992b)], (i) and (viii) in [Zalinescu (1998)] for X,Y normed spaces, while conditions (ii) and (vi) were introduced in [Zalinescu (1999)]. Condition (vii) of Theorem 2.8.3 was introduced by Borwein (1983) for Yo = Y (see also [Zalinescu (1986)]), (x) was introduced by Borwein and Lewis (1992), (ix) was introduced by Moussaoui and Voile (1997), (i) by Zalinescu (1998) for X, Y normed spaces, (ii) was introduced by Combari et al. (1999), (iv) was introduced by Zalinescu (1999) (a slightly stronger form was used in [Simons (1990)]) as well as conditions (v) and (vi). For C G L(X, Y), generally, Theorem 2.8.6 is obtained under the corresponding conditions of Theorem 2.8.3 (taking / = 0). For C a densely defined closed linear operator, Rockafellar (1974) and Hiriart-Urruty Hiriart-Urruty (1982) obtained the results for g lsc, X, Y Banach spaces and Yo = Y (in [Rockafellar (1974)] X is reflexive and e = 0); Aze (1994) obtained the formula for the conjugate under condition (ii) in normed spaces. For general C condition (iv) was used by Zalinescu (1992b); the other conditions (excepting (v)) were used in [Zalinescu (1999)]. Condition (ix) of Theorem 2.8.7 was used by Joly and Laurent (1971), (vii) for X a Banach space was introduced by Attouch and Brezis (1986), (vi) was introduced by Zalinescu (1992b), (i) was introduced by Aze (1994) in an equivalent form in normed spaces, while (ii) was introduced by Combari et al. (1999). Corollary 2.8.8 was obtained by Voile (1994) for X a Banach space, / , g lsc and '* replaced by !C. Formula (2.65) from Proposition 2.8.9 was obtained by Aze (1994) and Simons (1998b) for x = 0 € ( d o m / — domg) 1 when / , g are lsc. The case / = 0 of Theorem 2.8.10 was considered by several authors; condition (iii) was used in [Hiriart-Urruty (1982); Lemaire (1985); Zalinescu (1983a); Zalinescu (1984)]; the algebraic case was considered in [Kutateladze (1979a); Kutateladze (1979b); Zalinescu (1983a)]; Zalinescu (1984) considered a stronger version of (v) (for lsc functions and Yo = Y); note that in these papers g is assumed to be Q-increasing on the entire space. The general case was considered in [Combari et al. (1994)] under (iii) and a condition stronger than (v) (/, g, H lsc and lc instead of lb) and in [Combari et al. (1999)] under condition (i) and a variant of (iv); it was also considered by Moussaoui and Voile (1997) under condition (vii). The formula for d(p(x) in Corollary 2.8.11 can be found in [Combari et al. (1994)]. The formula for d
158

Convex analysis in locally convex spaces

Section 2.10: The minimax theorem is also classical and can be found in the books [Barbu and Precupanu (1986); Simons (1998a)]. Exercises: Exercise 2.2 is from [Penot and Bougeard (1988)], Exercise 2.3 is from [Crouzeix (1981)], Exercise 2.6 is from [Marti (1977)], Exercise 2.15 is from [Crouzeix et al. (1992)] (there in a more complete form), Exercises 2.18 and 2.29 are from [Hiriart-Urruty and Lemarechal (1993)], Exercise 2.19 is from [Lions and Rochet (1986)], Exercise 2.20 is the celebrated cancellation lemma from [Hormander (1955)], Exercise 2.21 is from [Hiriart-Urruty (1998)], Exercise 2.22 is from [Jourani (2000)], the assertions in Exercise 2.23 are well-known (see also [Aze (1997)]), Exercise 2.25 can be found in [Borwein and Kortezov (2001)] (in normed spaces), the formula for goo under condition (vii) of Theorem 2.8.1 in Exercise 2.26 is obtained in [Amara and Ciligot-Travain (1999)] and [Amara (1998)], Exercise 2.27 is the well-known Toland-Singer duality formula (see [Toland (1978)] and [Singer (1979)]), Exercise 2.30 is from [Zalinescu (1999)], the equivalence of a) and c) (for Banach spaces) in Exercise 2.32 can be found in [Bauschke et al. (2001)] (see also [Barbu and Da Prato (1985)]), Exercises 2.33 and 2.34 are from [Eremin and Astafiev (1976)], Exercises 2.35, 2.36 and 2.37 are from [Simons (1998a)], the last formula in Exercise 2.38 is from [Walkup and Wets (1967)], Exercise 2.39 is from [Penot (1998a)], the formula for the subdifferential of the function considered in Exercise 2.40 and its proof are from [Aussel et al. (1995)], the equivalences (e) •£> (f) and (g) & (h) of Exercise 2.41 are from [Zalinescu (1983b)], the equivalence of conditions (ii)-(iv) in Exercise 2.43 are established in [Adly et al. (2001b)] in reflexive Banach spaces, a weaker variant of the implication (a) => (c) of Exercise 2.44 is stated in [Aze and Rahmouni (1996)], Exercise 2.45(c) is from [Borwein and Vanderwerff (1995)], Exercises 2.47, 2.48 axe from [Laurent (1972)].

Chapter 3

Some Results and Applications of Convex Analysis in Normed Spaces

3.1

Further Fundamental Results in Convex Analysis

Throughout this chapter X, Y are normed spaces and X*,Y* are their duals endowed with the dual norms. As an application of Ekeland's variational principle and of some results relative to convex functions, we state the following multi-propose generalization of the Br0ndsted-Rockafellar theorem. Theorem 3.1.1 (Borwein) Let X be a Banach space and f £ r ( X ) . Consider e £ P, XQ £ d o m / , XQ £ def{xa) and /3 € K+. Then there exist xe £ X, y* € Ux- and Ae £ [—1, +1] such that \\xe - Xo\\ + P • \(X£ - X0,X*0)\ < y/E, x% := x*0 + VE(y* + /3\ex*) £ df{xe).

(3.1) (3.2)

Moreover ||* e - soil < Ve,

\te-xl\\
\(xe-x0,x*)\<e x*e£d2ef(x0),

+ P\\xl\\),

+ JilP,

\f(xe)-f(x0)\<e

(3.3) (3.4)

+ ^/fi,

(3.5)

with the convention 1/0 = oo. Proof.

The function

|| • Ho •. x - • K, \\x\\0~\\x\\

+

p-\(x,x*)\,

is, obviously, a norm on X, equivalent to the initial norm. Therefore {X, ||-||0) is a Banach space. Consider the function g := / — XQ. It is obvious 159

160

Some results and applications of convex analysis in normed spaces

that Xo £ dom
: g{x)>g(x0)-e

[& x*0 £

dsf{x0)].

We apply Ekeland's theorem (Theorem 1.4.1) to g, y/e and the metric d given by d(x,y) := \\x — y\\o. So there exists xe £ domg such that g{xE) + y/e- \\x£ -x0\\o \/x£X,

x^xe

< g(x0),

(3.6)

: g(x£) < g(x) + y/e • \\x - xe\\0.

(3.7)

From Eq. (3.6) we obtain that g(xe) + Ve(\\xs - ar0|| +0-\{xe-

x0, XQ)\) < g(x0) < g(xe) + e,

whence Eq. (3.1) follows immediately. Let us consider the function h : X -> E, h(x) := g(x)+y/e-\\x-xE\\0

-

f{x)-{x,xl)+y/E-\\x-xE\\+Py/e-\(x-xe,xl)\;

by Eq. (3.7) xe is a minimum point of h. Therefore 0 £ dh(xe). Since h is the sum of four convex functions, three of them being continuous, and taking into account the expression of the subdifferential of a norm (Corollary 2.4.16) and of the absolute value of a linear functional (at the end of Section 2.8), we have that 0 £ dh(xe) = df(x£)

- x* + y/i • Ux- + PVe • [ - 1 , +1] • x%.

Therefore there exists x* £ df(xs), y* £ Ux* and Xe £ [—1,1] such that Eq. (3.2) is verified. The estimations from Eq. (3.3) follow immediately from Eqs. (3.1) and (3.2). Using again Eqs. (3.1) and (3.2) we obtain that |(a;e -x0,x*

-XQ)\ = y/e-\{xE

- xQ,y* + p\Ex%}\

< v/i(||z e - x0\\ • W\

+ 0\Xe\ -\(xe-

< y/e(\\xs - x0\\ + fi • \{x£
x0,x*0)\)

-x0,x*0)\)

= e.

(3.8)

From Eqs. (3.1) and (3.8) we get |(a;e -x0,x*)\

< \{xe -x0,x*

-x^)\

+ \(x£

-X0,XQ)\


^IP,

Further fundamental

results in convex

analysis

i.e. the estimation in Eq. (3.4) holds, too. Since x$ G dEf(xo) df(xE), we get (X0 ~Xe,X*£

-XQ)

+ (X0 -X£,X*Q)

= (X0 ~Xe,X*) < (X0

< f(x0)

-XE,XQ)

161

and x* G

~f(Xe)

+£.

Using relations (3.1) and (3.8), we obtain that \f(x0) - f(xe)\

< \(x0-xe,x*0)\+s<e

+ VE//3,

i.e. the second relation in (3.5) holds. Using Eq. (3.8) and the fact that XQ G dEf(x0), x* G df(xE), we obtain that (x-X0,X*e)

= (X-X£,X*)

+ (Xe -X0,X*

-XQ)

+ (XE

-X0,XQ)

< f{x) - f(xe) +s + f(x£) - f{x0) + e - f(x) - f(x0) + 2e for every x G X, i.e. x£ G d2ef(x0); holds, too.

hence the first relation in Eq. (3.5) •

The next result is well-known. The first part is an immediate consequence of Borwein's theorem, while the density part, which follows easily from the first one, will be reinforced in Theorem 3.1.4 below. Theorem 3.1.2 (Br0ndsted-Rockafellar) Let X be a Banach space and f € r ( ^ ) - Consider e > 0 and (XO,XQ) G grd6f. Then there exists (x£,x*) G gr<9/ such that \\x£— XQ\\ < ^Jl and \\X*—XQ\\ < y/e. Inparticular domf C cl(dom<9/) and dom/* C cl(Im<9/). D Another consequence of Borwein's theorem is the following result which will be completed in Proposition 3.1.10 below. Proposition 3.1.3

Let X be a Banach space and f G T(X). Then

f(x) = s u p ^ z - z,z*) + f(z) I (z,z*) G grdf} = sup{(x, z*) - /•(**) | z" G lm(df)}

(3.9)

for every x G d o m / . Proof. The second equality in Eq. (3.9) is obvious because f(z)+f*(z*) = {z, z*) for any (z, z*) G grdf. Let x G dom / be fixed. When (z, z*) G grdf we have (a; — z, z*) + f(z) < f(x) for every i £ l ; hence f(x) > sup{(a; z,z*) + f(z) | [z,z*) egrdf}. Because / is lsc, by Theorem 2.4.4 (hi), there

162

Some results and applications of convex analysis in normed spaces

exists x* € de/2f(x). Applying Borwein's theorem we get {xe,x*) £ grdf such that x* 6 def(x). Therefore (xe — x,x*) < f{xe) - f(x) + e, whence f(x) — e < {x — xe,x*) + f(xe). Hence relation (3.9) holds. • As announced before, the density results in Theorem 3.1.2 can be stated in a stronger form. In the next result (xn) —>/ x means that (a;n) —> x and {f(xn)) -* f(x), and similarly for (xn) ->•/• x*. Theorem 3.1.4

Let X be a Banach space and f € r ( X ) . Then

(i) Vx e d o m / , 3((xn,xn))

Cgrdf

: (a;„) -+/ x;

(ii) Vx* E dom/*, 3 ( ( x „ , < ) ) C gvdf

: « ) - • , . x*.

Proof, (i) Consider x € d o m / . Because / is lsc, by Theorem 2.4.4 (hi), for every n S N there exists x* € dn-2f(x). Applying Borwein's 2 theorem for (x,x*), e = n~ and /? = 1, we get (xn,xn) € grdf such that \\xn — x\\ < n _ 1 , \f(xn) — f(x)\ < n~2 + n~x. Hence (xn) -^/ x. (ii) Because / * is lsc, it is sufficient to show that for every e > 0 there exists (y, y*) e df such that \\y* - x*\\ < e a n d / * ( y * ) < f*(x*)+e. Fixx £ d o m / . Let £> 0 and take r := \\x\\ + 2e-1 (e + f(x) + f*(x*) - (x,x*)) > 0. Consider also the function g := fO(x* + irux) € A(X). Then g(x) > (fnx*)(x) = (x,x*) - f*(x*) for every x € X. It follows that g € T(X) and f(x) + (—x,x*) > g~(0) > —f*(x*). By Proposition 3.1.3 there exists (z,z*) € dg such that -g*(z*) = g(z) - (z,z*) > -f*(x*) - e/2. But + (x* + trUx)*(z*) = f*(z*) + r l|ar* - z*\\. Therefore g*(z*) = f*(z*) f*(z*) +

r\\x*-z*\\
Because f*(z*) > (x,z*) - f{x) > - \\x\\ • \\z* - x*\\ + (x,x*) - f(x), from the preceding inequality we obtain that (r — \\x\\) \\z* — x*\\ < e + f(x) + f*(x*) — (x,x*), and so \\z* - x*\\ < e/2. The inequality above shows also that f*(z*) < f*(x*)+e/2. On the other hand, because (z,g~(z)) € cl(epi^), there exists (zn) C dom / and (un) C rUx such that (zn + un) —> z and limsup(/(z n ) + (u„,a;*»
(3.10)

we have that

0 < £„ = f{Zn) + (U„, X*) - (Zn + Un, Z*) + (Un, Z* - X*) +

f*(z*).

Taking into account that (un,z* — x*) < r \\x* — z*\\, from Eq. (3.10) we obtain that limsup£ n < 0. Hence lime„ = 0. Because (un) is bounded, so

Further fundamental results in convex analysis

163

is (zn). Therefore there exists r' > 0 such that (zn) c r'Ux- Take n E N so that S := en < 1 and VS(r' + 2)(1 + \\z*\\) < e/2; set z := zn. Of course, we have that z* G dsf(z). By Borwein's theorem (for /? = 1) there exists (y,y*) G df such that \\z - y\\ < y/S, \\z* - y*\\ < VS{1 + ||z*||) and |/(«) ~ f(v)\ <S + VS. Hence ||j/* - x*\\ < y/S{l + \\z*\\) + e/2 < e and /*(»*) = -/(V) = (y,y* - z*) + (y-z,z*)-6

+ f*(z*) + f{z) -

f(y)

< VS(V5 + r') (1 + ||z*||) + \f5 \\z*\\ -5 + f(x*) < f*(x*) + e/2 + VS(r' + 2)(1 + ||«*||) < f*(x*)

+s/2 + 6 + V6 + e.

The proof is complete.

•

Using Br0ndsted-Rockafellar's theorem we can add other sufficient conditions for the validity of the conclusions of Theorems 2.8.1 and 2.8.7. Proposition 3.1.5 Let X,Y be Banach spaces, F £ T(X x Y), A G L(X,Y), D = {Ax - y \ (x,y) € d o m F } and E = {Ax - y \ (x,y) € dom&F}. Then ic

E = iiE = ic(coE) = ri(coE) = icD = riD.

Moreover, if one of the above sets is nonempty then icE = E — D; in particular the sets ICE and E are convex. Proof. Consider the linear operator T : X xY ->• Y defined by T(x, y) := Ax — y. Since domdF C dom.F and d o m F is convex, we have that E = T(domdF)

CcoE

= T(co{domdF))

By Theorem 3.1.2 we have that d o m F C domdF. nuity of T, we have that

CD = T ( d o m F ) . Hence, using the conti-

E C D C T ( d o m d F ) C T(domdF) = E. Using the properties of the affine hull (see p. 2), it follows that aff E C aff D c aff E. Therefore, if aff E is closed then aff D — aff E; the above relation shows that %CE c %CD. Let us show the converse inclusion. Let 2/o G %CD and consider the function F 0 6 T(X x Y), F0(x,y) :— F(x,y — yo)Of course, domF 0 = d o m F + (0,j/o) and dom9F 0 = d o m 9 F + {0,y0). Therefore 0 € i c ( P r y ( d o m F 0 ) ) . Taking
164

Some results and applications of convex analysis in normed spaces

such that dF0(x,Ax) = dF(x,Ax — y0) ^ 0. Therefore yo € E. So we obtained that *D = ICD C E C D, which shows that aff E = aff-D and *D C iE. It follows that icE = icD = ic(coE). As observed after the proof of Theorem 2.8.1, in our situation, if %CD is nonempty we have that ic D = rintD. Suppose that %CD 9^ 0 (for example). From what was proved above, we have that rinti? = lD = icD = rintE = lE = icE C E C D C E. Hence E~ = T°E = D:. The conclusion follows.

•

Remark 3.1.1 Taking into account the preceding proposition, for X, Y Banach spaces and F G T(X x 7 ) , we may add to the sufficient conditions in Theorem 2.8.1 the following conditions: 0 G ri{Ax - y \ (x,y) £ ic

0 £ {Ax -y\(x,y)€ ic

0 € {Ax -y\(x,y)e

domdF}, domdF}, co(domdF)},

y 0 = cone{ Ar - y | (x, y) 6 dom OF} is a closed linear space. Indeed, the first three conditions are, evidently (using the preceding proposition), equivalent to 0 £ lc{Ax — y \ (x,y) £ d o m F } . In the fourth case y 0 = aff {Ax — y\(x,y)e co(dom OF)}, and so 0 e lc{Ax — y\(x,y)e domF}. An important particular case of the preceding proposition is when X = Y, A = Idx and F(x,y) = f(x) + g(y) with f,g 6 T(X); in this case D — d o m / — domg and E = domdf — domdg. Using the Borwein theorem one obtains a formula for the subdifferential of a composition g o A of a lsc convex function and a continuous linear operator without qualification conditions. Theorem 3.1.6 Let X,Y be Banach spaces, A e C{X,Y), g £ T(Y), f = g o A and x 6 d o m / . Then x* 6 df(x) if and only if there exists a net ((yi,yt))ieI C grds such that (yt) ->• y := Ax, (g(yi)) ->• g{y), ((yi-y,y*))^Oand(A*y*)^x*. If X is reflexive one can take sequences instead of nets and impose norm convergence instead of w* -convergence.

Further fundamental

Proof.

Let (fa,y*))i€l

results in convex

analysis

165

C gidg be such that fa) -> y, (fa - y,y*)) ->•

0, {gfa)) -> g(y) and (i4'i/r) ^ x*. Then (y - yhy*) i G I and t / S F . It follows that (x' - x,A*y*) + (y- yhy*)

= (Ax' - yuy*)

< gfa-gfa)

for all

< f(x') - gfa)

for alH G / and x' G X. Taking the limit we obtain that (x' — x,x*) < /(x') - f(x) for all x' G X, and so x* G d/(x). Let now x* G df(x) and consider (£„)„6N 4- 0- Using Corollary 2.6.5 we have that x* G w*-clA*(d£ng(y)) for every n G N. Let A/" be a base of w*neighborhoods of x* and consider / = Nx M with (n, V) y (n1, V) iff n > n' and V CV. Then for all i = (n, V) G / there exists z* G dEng(y) such that A*£* G V. Taking /? = 1 and e = e^ in Theorem 3.1.1, there exists (yi,y*) G 3g such that \\yi - T/|| < e„, ||y* - ^ | | < en, \g(yt) - g(y)\ < en(en + l) and \(Vi ~ V,V*)\ < en(en + 1). Because \\A*z* - A*y*\\ < \\A*\\ • \\y* - z*\\ < en \\A*\\ and (A*z*)ieI ^> x*, we obtain that (A*y*)i€l ^> x*. If X is reflexive then w*-cli*(9 fB g(j/)) = cl A* (dSng(y)), and so, for every n G N, we can take z*n G dEng(y) such that ||A*z* — x*|| < e„. The conclusion follows. • Using the preceding result one obtains a similar formula for the subdifferential of the sum of two lsc proper convex functions. Theorem 3.1.7 Let X be a Banach space, / i , / 2 G T(X) and x G dom / i fl dom / 2 . Then x* e S f / i + M (x) if and only if there exist two nets {(xk,i>xt,i))ieI C gr<9/fc, k = l,2, such that ( x M ) i e / ->• x, (fk(xk,i))ieI -> /fc(x), ( ( x M - x , x ^ i ) ) . e / -^0 for k = 1,2 and (x*^ + x*2x*. 7/X is reflexive one can take sequences instead of nets and impose norm convergence instead of w* -convergence. Proof. We apply the preceding result iorY ~ XxX, A: X -> Y defined by Ax := (x,x) and# : Y -> E defined by #(xi,x 2 ) := /i(xi)+/2(x2). Then / := /1+/2 = g°A. The sufficiency is immediate by taking y = (x, x) = Ax, Vi = (xi,i,X2,i) and y* ~ (xl^xlj) for i G i". Let x* G df(x). By Theorem 3.1.6 there exists a net ((yi,y*))ieI C gvdg verifying the conditions of the theorem with y := (x,x). Taking yt = (xi ) j,x 2 ,i) and y* = ( x ' ^ x ^ j ) and taking into account that dg(x\,X2) — dfi(x{) x 0f2(x2), we have that ((xk,i,x*ki))ieI C gvdfk and (xk,i)iei -»• x for k = 1,2. Assume that (fi(xi,i)) A fi(x)- Because /1 is lsc at x, limsup i 6 / /i(xi,i) > /i(x). Thus

166

Some results and applications of convex analysis in normed spaces

there exists S > 0 such that J := {i £ I \ fi(xiti) — fi(x) > 5} is cofinal. It follows that g(yi) - g(y) > 6 + (f2{x2,i) — f2{x)) for all i £ J. Taking the limit inferior, we obtain the contradiction 0 > S. Therefore {fk(%k,i))ieI -^ fk(x) for k — 1,2. The inequalities fi(xi,i)

- fi(x) < (xij -x,x{A)

for i £ I imply that ({xij — x,x{ti})

< (yi -y,y*)

- {h(x2,i)

-

f2(x))

-> 0.

D

Using Br0ndsted-Rockafellar theorem we obtain another famous result. Theorem 3.1.8 (Bishop-Phelps) Let X be a Banach space and C C X, C 7^ X, be a nonempty closed convex set. Then (i)

The set of support points of C is dense in B d C .

(ii) The set of support functionals of C is dense in the set of continuous linear functionals which are bounded above on C. Moreover, ifC is bounded, then the set of support functionals of C is dense in X*. Proof, (i) Let x0 6 Bd C and e £ ]0,1[. Taking into account that C ^ X, there exists xi £ X \C such that \\xi — x 0 || < £2- Applying a separation theorem, there exists XQ £ Sx> such that SVLPX£C{X,XQ) < (XI,XQ). SO, for every x £ C, {x - X0,XQ)

= (x - XI,XQ)

whence XQ £ de2tc(xo)have that

+ (xi - X0,XQ)

< (x - xi,xl)

+ \\x\ - x0\\

<

e2,

Applying the Br0ndsted-Rockafellar theorem we

3xe £ C, 3a;* £ dbc{xe)

: \\x£ - x0\\ < e, \\x* - XQ\\ < e < 1.

Since \\XQ\\ = 1, we have that x*e ^ 0, and so xE is a support point of C with \\x£ — xo\\ < e. The conclusion holds. (ii) Using the second part of Theorem 3.1.2 we have that dom(tc)* C cKJmdic) = cl(Imdt c \ {0}), which shows that the conclusion of the theorem is true.

•

The following theorem will be used for a simple proof of the Rockafellar theorem.

Further fundamental results in convex analysis

167

Theorem 3.1.9 (Simons) Let X be a Banach space, f € T(X) and x € X, n G R be such that inf / < 77 < f(x). Consider L :=

sup -fizz rr, x€X\{x} ||ar — ar||

and dx • X -V R, dj(x) := | | i — x||. Then: (i) 0 < -L < oo and inf (/ + Ldx) > n; (ii) V e e ] 0 , l [ , 3 i / 6 X

: ( / + L«fc)(i/) < i n f ( / + i d * ) + e L | | i - y\\;

(iii) V e € ] 0 , l [ , 3 («,*•) G g r d / and\\z*\\ < (l + s)L;

: {x - z,z*) > (I - e)L\\x - z\\ > 0

(iv) Ve €]0,1[, 3(z,z*) € g r S / : (z - z,z*> + /(*) > 17 and L < \\z*\\<(l + e)L. Proof, (i) Because 77 > inf / , it is clear that L > 0. Since 77 < /(a;) and / is lsc at x, there exists p > 0 such that / ( x ) > 77 for every a; € B(x,p). Furthermore, since / € r ( X ) , there exist x* € X* and a 6 R such that f(x) > (a;!a;*) ~ Q f° r every a; £ X. Therefore 77 — f(x) p. Let 7 := max{0,r\ + a — (x, x*)}. So,

V* £ X, \\x - x\\ > p : 5_jlZg) < ||x*|| + _ ] _ < li^H + 2. IF ~ x\\

\\x ~ x\\

P

This relation and the choice of p show that L < 00. From the expression of L we obtain immediately that inf (/ + Ldx) > 77. (ii) Let e e]0,1[. Since (1 - e)L < L, from the definition of L there exists y £ X, y ^ x, such that (77 — f{y))/\\x — y\\ > (1 — e)L, and so (/ + Ldx)(y) < 77 + eLdi(y) < inf(/ + Ldx) + eL\\y - x\\. (iii) Let e €]0,1[ be fixed. The function / + Ldx 1S proper, lsc and bounded from below, while the element y from (ii) is in dom(/ + Ldx) = d o m / . Taking the metric d o n X defined by d(xi,X2) := L\\xi—X2\\, (X,d) is a complete metric space. Using Ekeland's variational principle we get the existence of z € X such that (/ + Ldx)(z) + EL\\Z - 7,11 < (/ + Ldx)(y)

168

Some results and applications of convex analysis in normed spaces

and (f + Ldw)(z) < {f + Ld¥)(x) + eL\\x - z\\

VxeX.

The first relation and (ii) give ||^ — yj| < \\x — y\\, and so z ^ I . The second relation says that z is a minimum point of the function / + Ld^ + sLdz. Taking into account that cfe- and dz are continuous convex functions, it follows that 0 G d(f + Ldw + eLdz)(z) = df(z) + Ldd^{z) +

eLddz(z).

But ddz{z) = d\\ • ||(0) = Ux- and dd^(z) = {x* € Ux* | (z - x,x*) = \\z — x\\}. Hence there exist z* G df(z) and x*, y* £ Ux* such that z* = Lx* + sLy* and (x - z, x*) — \\x - z\\. Therefore ||z*|| < (1 + e)L and (x — z, z*) = {x — z, Lx* + eLy*) = L\\x — z\\ + eL(x — z, y*) > L\\x - z\\ - eL\\x - z\\ = L(l - e)\\x - z\\ > 0. (iv) Let e £]0,1[ be fixed and consider e' := e/3. Let us take M := (1 + 2e')L. Since / + Md% > f + Ld% > r], we can apply Ekeland's theorem for / + Md-x, an element xo of dom / , e' > 0 and the metric defined at (iii). We get so the existence of z e X such that V i e l :

(f + Mtk)(z)<(f

+

Mds)(x)+e'L\\x-z\\.

As in the proof of (iii), there exist z* € df(z), x*, y* € Ux* s u c h that z* = Mx* +e'Ly* and (x-z,x*) = \\x - z\\. Thus ||z*|| < (1 + s)L and (x - z, z*) > (M - e'L)\\x - z\\ = (1 + e'L)\\x - z\\. So, for x = z we have that (x — z, z*) + f(z) = f(x) > rj, while for ~x yi z we have (x-z,z*)

+ f(z) > (l + e')L\\x-z\\

+ f(z) = (f + Ldw)(z)+e'L\\x-z\\

> v.

Therefore (x — z, z*) + f(z) > T). Moreover ||i - z|| • \\z*\\ >(x-x,z*) >V~

= [(x - z, z*) + f(z)] - [(x - z, z*) + f(z)]

f(x)

for every x € X; the last inequality holds because z* G df(z). ||af — x\\ for x 7^ x, we obtain that ||z*|| > L.

Dividing by D

Convexity and monotonicity

of

subdifferentials

169

Using the preceding theorem we reinforce slightly Proposition 3.1.3. Proposition 3.1.10 Let X be a Banach space and f 6 T(X). relation (3.9) holds for every x 6 X.

Then

Proof. Taking into account Proposition 3.1.3 we have to show that oo = sup{(a; — z,z*) + f(z) \ (z,z*) € gr<9/} when x £ d o m / . So, let x £ X \ d o m / and take inf / < 77 < 00. Using Theorem 3.1.9 (iv), there exists {z,z*) E gr<9/ such that (a; - z,z*) + f(z) > rj. The conclusion follows. • The maximal monotone operators are of a great importance in the theory of evolution equations. A significant example of such operators is the subdifferential of a proper lsc convex function on a Banach space. Theorem 3.1.11 (Rockafellar) Let X be a Banach space and f 6 T(X). Then df is a maximal monotone operator. Proof. Let (x,x*) EXxX*\gvdf. Then x* £ df{x), and so 0 i df(x), where / := / — x*. It follows that inf/ < fix). Applying assertion (iii) of Simons' theorem, there exists (z, z*) € g r 9 / such that (x — z,z*} > 0. Consider z* := z* + x*\ we have that (z, z*) £ df and (z -x,z*— x*) < 0, which shows that the set df U {(x,x*)} is not monotone. Therefore df is a maximal monotone operator. • 3.2

Convexity and Monotonicity of Subdifferentials

The aim of this section is to show that the monotonicity of an abstract subdifferential of a lower semicontinuous function ensures its convexity; among these abstract subdifferentials one can mention the Clarke subdifferential we introduce below. Throughout this section (X, ||.||) is a normed vector space. Consider | / M c I and x e clM. In the sequel we shall denote by (x n ) - > M X a sequence (xn) C M with (xn) -» x; more generally, x - > M X will mean x £ M and x —> x. The Clarke's tangent cone of M at x is defined by Tc{M,x):={ueX\V{tn)

10, V(xn)-*Mx,

3(un)^u, V n e N : xn + tnun 6 M).

Recall another cone introduced in Section 2.3: C(M,x) := e l f | J

t-(M-x))

=cone{M-x).

170

Some results and applications of convex analysis in normed spaces

Because clM -x C C(M,x) Note (Exercise!) that

C C(clM,i), we have C{M,x) = C(clM,:r).

0ETc(M,x)cC(M,x).

(3.11)

Several properties of the tangent cone in the sense of Clarke are collected in the following proposition. Proposition 3.2.1 (i) Tc{M,x)

Let 0 ^ M C X and xEc\M.

Then:

is a nonempty closed convex cone;

(ii) TC(M, x) = Tc(cl M, x) = Tc(M D V, x) for all V E M{x); (iii)

if M is a convex set then Tc(M,x)

=

C(M,x).

Proof, (i) It is obvious that Xu E Tc(M,x) for A > 0 and u E Tc(M,x); hence Tc(M,x) is a cone. Let u,v € Tc(M,x) and consider (tn) 4- 0 and (xn) ~>M x. Then there exists (un) -> u such that x'n := xn + tnun G M for every n E N. Of course (xJJ —>• 3;. Hence there exists (vn) —> u such that £n + in(w„ + u„) = x^ + i n w n G M for every n. Therefore u + v E Tc(M,x). Consider now (uk)ke^ C Tc(M,x) with (ufc) -+ u E X. Let us show that u E Tc{M,x). For this take (tn) I 0 and (xn) - > M £• Because ufc E Tc{M,x), there exists (u*) ->• ufc such that xn + tnukn E M for every n G N. Hence xn+tnukn E M for all k, n E N. For every n EN there exists k'n E N such that ||u* — uk\\ < n~l for every k > k'n. Let (fc„) C N be an increasing sequence such that kn > k'n; then ||u* - uk\\ < n _ 1 for all k,n E N with k > A;„. Let u„ := u* n . Of course, £„ + £ n u„ = xn + tnuknn G M for every n. As ||M„ - u|| = \\unn - u|| < ||u*n - ukn || + \\ukn - u\\ < n _ 1 + ||u*n - u||, we have that (un) ->• u. Hence u G Tc(M,x). Therefore Tc(M,x) is a closed convex cone. (ii) Let u E Tc(M,x) and take (tn) I 0 and (xn) - > C I M £• For every n G N there exists xn E M such that \\xn — xn\\ < tn/n, i.e. xn = xn + tnn~1u'n with u'n E Ux- Hence (xn) ->M x. Therefore there exists («„) -> u with xn + tnun = xn + tn(n~lu'n + un) G M C clM for every n E N. As (n _ 1 u^ + u„) —> u, we have that u G I b ( c l M , x ) . Conversely, let u G Tc(c\M,x) and take (tn) I 0 and (z n ) -tM x. There exists (un) -> u with xn + tnun G cl M for every n. Like above, for every n E N there exists u^ G Ux such that a;„ + tnun + tnn~lu'n G M . Because ( u „ + n _ 1 u ^ ) —> ii, we have that w E TQ^M^ X). Hence TQ^M, X) — Tc(dM,x).

Convexity and monotonicity

of

subdifferentials

171

The equality Tc{M,x) = 7 b ( M n V,x) is obvious when V is a neighborhood of x. (hi) Let M be convex. Taking into consideration (ii) we may assume that M is closed. Consider x E M and take (£„) 4- 0 and (xn) —»M X. There exists no € N such that tn < 1/2 for n > no. Take un := x + x - 2xn for n > n0 and un = 0 otherwise. Of course, (un) -¥ x — x. As xn + tnun = (1 — 2tn)xn + tnx + tnx S M for n > no, we have that x-x € Tc{M,x). It follows that C(M,x) C T c ( M , x ) . Using Eq. (3.11) we obtain the conclusion. D From (ii) and Eq. (3.11) we obtain that Tc(M,x)

Cn '

C(MHV,x)

=:

TB{M,x);

'V'gJV(x)

is the well known tangent cone in the sense of Bouligand of M atxed M. Let now / : X —> M. be a proper function and x 6 dom / . It is natural to consider the tangent cone Tc (epi/, (x, f(x))). This cone is related to the Clarke—Rockafellar directional derivative of / at x introduced as follows: TB{M,X)

,*,_ . f'{x,u):=sup

, limsup e>0 tiQ,(x,a)->epi,(x,f(x))

. f := sup ml

. „ mi

fix + tv) — a

ll«-«ll<e

*

sup

e>0 *>0 0
f(x +

ml

f(x)
II"-'"ll<

£

tv)-a

.

*

It follows that , ,_ , . , / ' t( a ; , w ) = s u p inf

sup

f{x + tv) -

mi

lk-"ll<E

£ > 0 <5>0 o
f(x)

,

*

and even /t(x,u)=sup

limsup

inf

e>0 (4.0, x->/5 ll«-«ll<e

/ (

*+

tV

} ~~f{x)

(3.12)

'

if / is lower semicontinuous at x, where x —>/ x~ means x —>• x and f(x) —>• f(x). It is obvious that / t ( x , 0) < 0 and /''"(xjAu) = A/ 1 '(x, u) for A > 0 and u G X. We may have f^(x, 0) = — oo even if / is continuous at x. Take for example / : R -»• E, f(x) := — \/\x\, and x = 0 (Exercise!). The next result shows the connection between the preceding two notions.

172

Some results and applications of convex analysis in normed spaces

Proposition 3.2.2 Let f : X —> R be a proper function and x G d o m / . Then epi ft(x, •) = Tc (epi / , (x, f(x))). In particular ft(x, •) is a Isc convex function; ft(x, •) is a Isc sublinear function if and only if ft(x, 0) = 0. Moreover, if f is convex then epi ft(x, •) = cl (epi f'(x, •)). Proof. Let (u,A) G Tc (epi f, (x, f(x))); assume that ft(x,u) > A, and take ft(x, u) > A' > A. Then there exist EQ > 0, ((xn,an)) -»epi/ (x,f(x)) and (tn) 4- 0 such that inf„ 6 c( U)£o ) t~* (f(xn + tnv) — an) > A'. Because (u, A) 6 Tc (epi f, (x, f(x))), there exists the sequence ((un,Xn)) converging to (u, A) such that (xn,an) + tn(un,\n) G e p i / , i.e. f(xn + tnun) < &n + tnXn for all n G N. Since (un) -» u, there exists no such that un G D(u,eo) for n > no- Hence

v^

-r

f(xn + tnv)-an

f(xn +

tnun)-an

A <

inf — < — < A„ Vn > n 0 . ||w—u||<e0 tn tn Taking the limit we get the contradiction A' < A. Let now f^(x~,u) < A and take ((xn,an)) —>epif (x,f(x)) and (tn) I 0. Let us fix (ek) I 0. Because • infr

5>0

• inft

sup 0
\\x-x\\<S, \a-f(x)\<S

f(x + tv) - a —
\\v-u\\<ek

t

for every k G N, there exists 8k > 0 such that . inf

sup 0«<<5fc,||x-x||<(5 fc ,/(x)
u

f(x + tv) -a — £

ll»- ll< *

. < A.

*

There exists n'k G N such that 0 < t„ < 8k, \\xn — x\\ < 8k, \an — f(x)\ < 8k for all n > n'k. Therefore inf„€£>(U)efc) t" 1 (f(xn + tnv) - an) < A, which shows that for every k and every n>n'k there exists u* 6 D(u, e^) such that f(xn + tnUn) < a „ + Xtn. Consider an increasing sequence (n^) C N such that nk > n'k for every k G N. Taking un := u* for n^ (u, A) and (x„,Q!n) + £„(u„,A) G e p i / for every n G N. Hence (w,A) G Tc (epi / , (x,f(x))). Because Tc (epi / , (x,f(x))) is closed, we have that epi ft (x, •) C Tc (epi f,(x, f(x))). Therefore epi ft(x,-) = Tc(epif,(x,f(x))). From this relation and Proposition 3.2.1 (i) we have that ft(x, •) is Isc and convex. The other statement follows from Proposition 2.2.7. Assume now that / is convex. Because cone ( e p i / - (x, f(x)))

C epi f'(x, •) C cone(epi/ - (x,

f(x))),

Convexity and monotonicity

we have that epip(x,

of

subdifferentials

•) = C (epi/, (x,f(x)))

173

= cl(epi/'(:r, •))•

D

+

Note that when / is not lsc at x, f^(x, •) and / (x, •) may be different, where, as usual, / is the lower semicontinuous envelope of / . Take for example / : R ->• R, f(x) = 0 for x ^ 0, /(0) = 1; / t ( 0 , •) = - c o , but / (0, •) = 0. When / is lsc at x then f*(x, •) and / (x, •) coincide. When / has additional properties, Z1" has a simpler expression. Proposition 3.2.3 Let f : X —> R be a proper function and x 6 d o m / . Then for every u € X we have: (i) limsupj.^. xP(x,u) ,-lv- N / f'(x,u)<sup

< f^(x,u);

• r inf

moreover, if f is lsc atx • inff

sup

f(x + tv)-

£>0 S>0 |j^—^||<<S,0
< inf

then

f(x)

*

sup

.

*>° \\z-x\\<S,0
(3.13)

*

(ii) / / / is continuous at x then

,-iv- x f'{x,u)—

•r

. t f{y + tv) - f(x)

sup inf

sup

mi

e>0 <5>0 \\x-x\\<S,0
.

ll"-ull<e

*

(hi) If f is finite and L-Lipschitz on B(x,r) for some r > 0 and L > 0, then ttr\ • f(x + tu) - / ( : r ) /'(a;,u) = inf( sup s>0

||x-x||<(5,0
t

f(x + tu) — f(x) , ,. .. = limsup^—-—'-— J -±-L < L -\\u\\. (iv) If f is finite and Gateaux differentiate tinuous at x then f^(x, •) = V/(3;).

on B(x,r),

(3.14)

and V / is con-

Proof. Throughout the proof u € X is a fixed element. (i) Let f*(x, u) < X and consider e > 0; there exists 6 > 0 such that sup \\x-x\\<8,0
. inf lk-«ll<

f(x + tv)-f(x) e

^ < A.

*

Take 6' := 6/2. Let x' e £(x,<5') with \f{x') - f(x)\ < 6' and x, t be such that \\x - x'|| <6',0
174

Some results and applications of convex analysis in normed spaces

0 < t < 8 and f(x) < f(x) + 8. It follows that sup {mt^D^r1

(f{x + tv) - f(x))

| t G ]0, 8'}, x e D{x', 8'), f(x)
l

< sup {miv€D(u>e)r

(f(x + tv) - f{x)) \te]0,8],

x £

D(x,8),

f(x) < f(x) + 5} < A for all x' e B(x,8') with \f(x') - f(x)\ < 8'. Therefore f^(x',u) < A for such x', and so lizn supx^. w f^(x,u) < f^(x,u). The first inequality in Eq. (3.13) follows immediately from Eq. (3.12), while the second is obvious. (ii) Taking into account Eq. (3.13), one must show the inequality >. For this take e > 0 and 8 > 0. There exists 8' € ]0,8] such that f(x) < f(x) + 8 for every x € D(x,8'). Then sup 0<£<<5 \\x~x\\
inf

^

H

llu-ull^e

M

>

sup

t

inf u

0
^

e

+

^ ~ ^ , t

whence the conclusion follows. (iii) The inequality < is proved in Eq. (3.13). Assume that / is Lipschitz on B(x,r) with Lipschitz constant L. Take A > lim sup x _,. s ^ 0 Hx+tu)~Hx) and fix e > 0. Consider 80 > 0 such that sup|| ;c _j||< 5o)0
<

f(x + tv) - f(x + tu)


f{x +

tu)-f{x)

+ tU

] ~ fix) < / ( * + tU] ~ ™ + Le.

Hence sup \\x-x\\<5 0
. , inf ll"-«ll<£

fix + tv) - fix) ^ '-—i±-± < t

sup \]x-x\\<6 0
fix + tu) - i fix) — — - ^ - L + Le, t

Convexity and monotonicity

of

subdifferentials

175

and so . mi

sup

inf

*>° ||i-i||<<5,0
<

£

sup

/(a; + fa) - / ( a ; ) — t J fix + tu)J - L f(x) -^——-— -^+Le<\

||a:-x||<<So,0«<<5o

+ Le.

*

Taking the limit for e -> 0, we obtain \ ^> rhm inf -t A

sup

e4-0 d>0 \\x-x\\<6,0
f(x + tv)-f(x) ^—L = ttf—f ] { x , u\) .

-t inf ll"-«ll<e

t

Taking into account (ii), we get Eq. (3.14). Taking 5 = r / ( l + ||w||), for x € B(x,S) and t £]0,S] we have that x + tu,x € B(x,r), and so f{x + tu) — f(x) < tL \\u\\, whence f^(x,u) < L \\u\\. (iv) Let r > 0 be such that / is Gateaux differentiable on B(x, r). Since V / is continuous at x, V / is bounded on a neighborhood of x. So we may assume that ||V/(x)|| < L for x € B(x,r). Using the mean-value theorem we obtain that / is i-Lipschitz on B(x,r), and so Eq. (3.14) holds. Let (tn) i 0 and (xn) -»• x be such that f^(x,u) = \imt~1 (f(xn + tnu) — f(xn)). But xn+tnu, xn £ 5(3;, r) for n > no (for some no £ N); applying the mean value theorem, there exists 6n £]0,i„[ such that f(xn + tnu) — f(xn) — V / ( x n + 6nu)(tnu) for every n > no- Because V / is continuous at x, we obtain that ft(x,u) = Vf(x)(u). O Let us introduce now the Clarke subdifFerential of / : X —> K at xeX with f(x) £ E. This is dcf(x)

= {x* € X* | {u,x*) < f\x,u)

Vu € X } ;

if f(x) £ E we consider that dcf(x) = 0. Taking into account Theorem 2.4.14, dcf(x) ^ 0 if and only if /^(af, 0) = 0. Moreover, dcf(x) is a nonempty w*-compact subset of X* if / is Lipschitz on a neighborhood of x. Other properties of dc are collected in the following result. Theorem 3.2.4 domg.

Let f,g : X —> R be proper functions and x £ d o m / n

(i) If / and g coincide on a neighborhood ofx then dcf(x) (ii) If f is convex then dcf(x) (iii)

=

df(x).

If x is a local minimum of f then 0 €

dcf(x).

— dcg(x).

176

Some results and applications of convex analysis in normed spaces

(iv) \imsupx_>x dcf(x)

C dcf(x),

where kmsupx^

dcf(x)

is the set

{x* G X* | 3 (xn) -+f x, 3 « ) % x*, Vn G N : < G 3 c / ( a ; n ) } • (3.15) (v) If g is finite and Lipschitz on a neighborhood of x then

(f + g)Hx, •) < fHx, •) + g\x, •), dcU + g)(?) c dcf(x) + dcg(x). (vi) / / g is Gateaux differentiate tinuous at x then

on a neighborhood of x and Vg is con-

(f + g)H*, •) = fHx, •) + V5(af),

dc(f

+ g)(x) = dcf(x)

+ Vg(x).

Proof, (i) and (iii) are obvious because p{x, •) = (flv^ix, •) for every V G N(x), while (ii) follows from the second part of Proposition 3.2.2. (iv) Let x* G limsupx^ fX-dcf(y); t n e n there exist (xn) ->/ x and (a£) ^ x* such that x*n G dcf(xn) for all n G N. Let M £ I be fixed. Then (u,a;*) < p{xn,u) for every n G N, whence, by Proposition 3.2.3 (i), (u,x*) < ft(x,u), and so x* G dcf(x). (v) Let r > 0 be such that g is L-Lipschitz on B(x, r); we may assume that L > 1. Let u £ X, e > 0 and 5 > 0 be fixed and take 6' = mm{S/{2L),r/(l + e + \\u\\)} > 0. Let v, x and t be such that \\v-u\\ < e , \\x-x\\ <5',0
+ g)(x) ~

f{x + tv) - f{x) t

g{x + tu) - g{x) t

whence, taking first the supremum with respect to v G D(u,e), we obtain that a < Ai(5) + A2(5) + Le, where a :=

sup ||a!-s||
.

mi H"-«ll<£

(f + g)(x + tv)-(f

+ g)(x)

t

U+9)(x)<(f+gm+s' Ai(S)

:=

sup \\x-x\\<S,0
inf — Il«-«II<£

A2(d) := sup {t'1 (g(x + tu) - g{x)) \x

eD(x,5)}.

—-^-L, t

,

Convexity and monotonicity

of

subdifferentials

177

Therefore /3 < Ai(S) + A2{5) + Le for every 5 > 0, where /3:=inf 5 >0 '

sup ||a!-»||<
inf if + 9)(x+ tv) - jf + g)(X) ^ £ Il -"II< * u

(f+9)(x)<(f+g)(x)+S'

Taking the limit of Ai (S) + A2 (5) for S —> 0 (taking into account that A\ and A2 are nondecreasing for 5 > 0), we get inf

sup

inf

(/ + g)(* + * 0 - ( / + g)(*)

5

>°\\x-x\\<5,0
•

f

< inf s>0

t

sup \\x-x\\<6,0
X

f(

f

inf ll"-"ll<

—£

+

tv

)

~

/(*)

-

—L

*

. 9(x + tu) — g(x) + inf sup — '-—^^- + Le. <5 >°||x-x||<(5 t

Taking now the limit for e —> 0, we get the desired conclusion. The relation for the subdifferentials follows from the preceding inequality and the fact that g^(x, •) is continuous. (vi) As mentioned in the proof of Proposition 3.2.3 (iv), g is Lipschitz on a neighborhood of x and so the conclusion of (iv) holds with g^(x, •) = Vg(x). Applying again (iv) for f + g and — g (we may assume that g is finite on X; otherwise take g = 0 outside a neighborhood of x), we obtain that f*(x, •) < (f + g)^(x, •) — Vg(x). The conclusion is now obvious. • In the rest of this section we use an abstract subdifferential. Before introducing this notion let us consider the following class of finite-valued convex functions: C(X) := {g : X -> E | g is convex and Lipschitz}; of course, €(X) is a convex cone in the vector space Rx of all functions from X into E. Let d ( X ) C E x be the cone generated by X*U{d [a , 6] | a, b € X) (C €{X)) and €2{X) C E x be the cone generated by

X*u{dfaM\a,beX}WD0, where

Do := { ^2k>QVkdlk

(Hk) C K+, ^2k>Ql*k

These cones will be used below.

=

*' (u*)*>°

conver

Sent} •

178

Some results and applications of convex analysis in normed spaces

We call an abstract subdifferential on the nonempty class T C M. a multifunction d : X x T =4 X*, which associates to (x,f) a set denoted by df(x), satisfying condition (PI) below: (PI) 0 G limsup y _> x d/(y) + dg(x) if / € T, g € <£(X), f(x) £ 1 and a; is a local minimum of / + g, where limsup _yX df{y) is defined as in Eq. (3.15) and dg(x) is the Fenchel subdifferential of g at x. A stronger form of (PI) is (P2) 0 € ~8f(x) + dg(x) if / 6 T, g e €(X), f(x) 6 1 and a; is a local minimum of / + g. Sometimes one asks (P3) df(x) = df(x) if / e

Tn€(X).

Of course, (P2) holds if (P4) and (P5) below are satisfied: (P4) 0 6 df (x) if / 6 J-, f(x) € E and x is a local minimum of / , (P5) T + €(X) C T and 8{f + g)(x) C df(x) + dg(x) when / G T and 9 e €(X). Note that WX + €(X) = RX,

A(X) + €(X) = A(X),

T(X) + €(X) = T(X).

Remark 3.2.1 From the preceding theorem we observe that Clarke's subdifferential dn and the Fenchel subdifferential d are abstract subdifferentials x on R and A(X), respectively; in fact they satisfy conditions (P1)-(P5). There are many other subdifferentials which satisfy condition (PI). Remark 3.2.2 If the abstract subdifferential d on T satisfies the stronger condition (P2) then for any proper function / 6 T one has df(x) C df(x) for all x € d o m / . Indeed, if x* 6 df(x) then a; is a (local) minimum point of / + (—x*), and so, by (P2), 0 £ df(x) + d(-x*)(x) = df(x) - x*. Hence x* € df(x). The following approximate mean value theorem proves to be useful in nonconvex analysis.

Convexity and monotonicity

of

subdifferentials

179

Theorem 3.2.5

(Zagrodny) Let (X, ||-||) be a Banach space and d be x an abstract subdifferential on T C E . Let f 6 T be Isc, a,b G X with a G d o m / and a ^ b, and r 6 l with r < f(b). Then there exist (xn) —>•/ c € [a, b[ and x*n G df(xn) for every n G N such that (i) r — f(a) < lim inf (6 - a, x*n), (ii) 0 < lim inf (c (iii)

xn,x*n),

f^(r-/(a))
(iv) ||& - a|| (/(c) - /(a)) < ||c - a|| (r - / ( a ) ) . Proof. There exists x* G -X"* such that (6 - a, x*) = r — f(a). Consider h := / — x*; then /i(a) < h(b). Because h is Isc, there exists c G [a,b[ such that h(c) < h(x) for all a; G [a,b]. Therefore c = (1 — /x)a + fib for some /i G [0,1[. It follows that c — a = fi(b — a) and ||c - a|| = fi \\b — a\\. Therefore /(c) - f(a) = h(c) - h(a) + (c — a,x*) < fi(r - / ( a ) ) , whence (iv) follows. Let 7 < h(c); then there exists r > 0 such that 7 < h(x) for every x G [a, b]+rUx- Otherwise, for every n G N there exists xn G [a, b ] + n - 1 { / x such that h(xn) < 7. Hence arn = dn + yn with d„ G [a, 6] and y n G n~1UxAs the segment [a,b] is a compact set, there exists a subsequence (dnk) converging to d G [a,b]. It follows that (xnh) —> d, and so, by the lower semicontinuity of h, we get the contradiction h(d) < 7. Let r > 0 correspond to 7 := h(c) — 1, and take C/ := [a, b] + rUx', of course, U is closed. Like above, for any n G N there exists r„ G ]0, r[ such that /i(z) > h(c) — n~2 for 3; G [a,b] + rnUx', choose tn > n such that 7 + tnrn > h(c) — n~2. Then one has Vz G U : /i(c) < h(x) + tnd[atb](x)

+ n~2.

(3.16)

Indeed, the inequality is obvious for x G [a, b] + rnUx- If x £ U \ ([a, b] + r„[/x) then d[0i6](x) > r„, and so /i(x) + t„d[a)ft](a;) > 7 + i n r n > ft.(c)-n-2. Consider i?„ — h + iu + *nd[0)(,]. Prom Eq. (3.16) we have that Hn(c) < infx Hn + n - 2 ; moreover, Jf„ is Isc and bounded from below. Applying Corollary 1.4.2 for Hn, c, e ~ n~2 and A := n _ 1 , we get un G X such that Hn(un)

< Hn(c),

||c-u„|| < n_1,

(3-17)

1

(3.18)

Hn{un) < Hn(x) + n - ||z - u n ||

Va; G X.

Since (un) —> c G [a, 6[, we may assume that un G intf/ for every n.

180

Some results and applications of convex analysis in normed spaces

Relation (3.18) can be written as ( / + tnd[atb] + n~ldUn - x*) (u„) < ( / + tnd[aib] + n - 1 d „ „ - x*) (x) for every x 6 U, where da denotes d[a,a] = ^{o}- Therefore un is a local minimum of/ + (tnd[a,b] +n~1dUn -x*); the function between the parentheses being convex and Lipschitz, by (PI) we have that 0 6 limsup <9/(2/) + d(tnd[a
n-Hl).

Let yn £ [a, b] be such that d{a^{un) = \\un — yn\\ (such an element exists because [a, b] is compact). Because 0, and so (yn) -> c. Since c ^ b, we may suppose that yn ^ b for every n 6 N. Hence yn = (1-A n )a+A„6 with An € [0,1[. From Proposition 3.8.3(H) on page 239, dd[a,t](u„) = d||-|| (u n -y n )r\N([a,b],y n ). Therefore ((1 - A)o + \b- yn,v*n) = (A - A„) (6 - a , < ) < 0 for every A € [0,1[. Hence (b — a, u*) < 0 for every n S N. Moreover, (c - u n , u*) = (c - y n ,v*) - (u„ - yn,v*) It follows that (b-a,u*n) n_11|& — a\\, whence

> (b-a,x*)

< - ||u n - y„|| < 0.

- ^(fr-a,^)

> r - f(a)

-

liminf (b — a,u*) > r — / ( a ) . Similarly, {c — un,u*n) > (c — u„,x*) — n _ 1 | | c - u n | | , whence liminf (c — un,u*n) > 0. From Eq. (3.17) we get f(un) - (un,x*) < /(c) - {c,x*), and so /(c) < liminf f(un) < l i m s u p / ( u n ) < /(c); hence l i m / ( u n ) = /(c) which shows that (un) —>f c. Because u*k e l i m s u p . ^ ^ df(u), for every k £ N there exist (u„,*) ->•/ Ufc and (u*^) ^> u£ (for n -» oo) such that u*nk G df(un>k)

for all n, A; e N.

Convexity and monotonicity

of

subdifferentials

181

It follows that ((c - uniu,u*nk)) -> ( c - Ufc,u£) for n -> oo. Therefore for every A: € N there exists n'fc € N such that IK,*-Wfc|| < fc_1, \f(un,k)(c-unik,u*nik)-

f(uk)\ < k'1,

\(b-a,u*nk

(c-uk,u*k)\

- u*k)\ < AT1,

< k~l

for n > n'k. Consider (nk) C N an increasing sequence such that nk > n'k for all k e N. Let xn := u„,fc and a;* := u^ fc for nk < n < nk+i- Since for nk k

~ Uk\\ + \\uk - C\\ < k~l

+ \\uk - C\\ ,

l / W - Z t o l ^ + l/toWtol, (b - a, < ) = {b-a, u*k) + {b-a, < < f c - u*k) > (b - a, u*k) (c-xn,x*n)

= (c-u n , f c ,u* ) A .) > (c-uk,u*k)

k'1

- fc-1,

we obtain that (xn) —•/ c, lim inf (6 — a, xn) > lim inf (b — a, u£) >r — f(a) n—>oo

k—•oo

and lim inf (c — x n ,a;*) > lim inf (c — Ufc,u^) > 0. n—s-oo

fc-+oo

Taking into account that b — c = (1 — /j,)(b — a) and \\b — c|| = (1 — /i) ||b — o||, from the preceding two relations we get lim inf (b- xn,x*n) = lim inf ((1 - fi) (b- a, a:*) + {c-

xn,x*n))

> (1 — /x) lim inf (b — a,xn) + lim inf (c — a;„,a;*) > K The proof is complete.

(r - / ( a ) ) . •

Remark 3.2.3 The conclusion of Theorem 3.2.5 remains valid if (PI) holds only for functions g in the cone £i(X) or in the cone C2PO defined on page 177. Indeed, for the first part just note that in the proof of Theorem 3.2.5 we used (PI) only for g = i„d[aj6j + n~1dUn — x* € £i(X). For the second part one must notice that one can find tn > 0 such that 7 + tnrn > h(c), and so Eq. (3.16) holds with d[Q>(,] replaced by d?a bi. Applying the Borwein-Preiss variational principle for p = 2, Hn,

182

Some results and applications of convex analysis in normed spaces

c, e := 2n~2 and A := \/2/n, we find un € B(c, y/2/n) and a function 0 2 ,„ generated by the sequences (fJ.n,k)k>o C K+ with X^>0Mn,ib = 1 and (wn,fc)fc>o C B(c, \/2) (see Eq. (1.11) on page 31) such that un is a local minimum of / + (tndfa M + n~1@2,n — x*). Then, in the expression of un we have tn \\un — yn || instead of tn and 6* 6 23/2Ux* instead of &£ 6 C/x* (see also Exercise 2.40). The rest of the proof is the same. Corollary 3.2.6

Let X be a Banach space and d be an abstract subx differential on T C M. . If f G T is a Isc proper function then for every x e d o m / there exists ((xn,xn)) C gr<9/ such that (xn) —»/ x and liminf (x — xn,x^) > 0. In particular d o m / C cl(domd/). Proof. Let x S d o m / . If z is a local minimum of / then 0 belongs to limsupy^ xdf(y) by (P2) (just take g = 0). Therefore there exists (xn) —>f x and (a;*) —> 0 such that a:* 6 df(xn) for every n € N. Of course, lim (x — xn, x^) = 0 in this case, and so the conclusion holds. Assume that x is not a local minimum of / . Then there exists (xn) —> x such that f(xn) < f{x) for every n £ N. Since / is lsc, we get (xn) -»•/ x. Applying the preceding theorem for xn, x and r = f(x), there exists ((xk,n,x%tn))keN C domdf such that (xk,n) ->/ yn G [x„,a;[ for k -» oo and 0

< |jx-lj (/W

_

/(^«)) < Hminf *,_>«, (a; - a;fc>„,a;^„),

/(l/n) < /(*„) + | ^ f (/(*) - /(*»)) < /(*) for every n € N. Since \\yn - x\\ < \\xn - x\\ we have that (yn) -> x, which, together with the preceding inequality and the lower semicontinuity of / , yields (yn) ->•/ x. Then, for every n 6 N there exists k'n € N such that lk*,n - Vn\\ < n _ 1 , |/(a:fc,„) - f(yn)\ < n _ 1 and 0 < (x - Xk,n,x*kn) f o r every k > k'n. Let (fc„) C N be an increasing sequence such that kn > k'n for n € N. Then lkfc„,n - z|| < ||a;*„,n - y n || + \\yn - x\\ < n _ 1 + \\yn - x\\, l/(**»,») - f(x)\

< \f{xkn,n)

- f(yn)\

+ \f(Vn) -

f(x)\

< n - 1 + |/(y„) - /(a:)| , 0 < (z-Z*„,„,4„,n) for all n £ N. Taking xn := Xkn,n and a;* := a;^ n , we have the desired conclusion. •

Convexity and monotonicity

of

subdifferentials

183

When T = A(X) and d is the Fenchel subdifferential, the statement of the preceding corollary is very close to assertion (i) of Theorem 3.1.2. Theorem 3.2.7 X

Let X be a Banach space, d be an abstract subdifferential

on T C K and f £ T be a Isc proper function. If df is a monotone multifunction then f is convex. Proof. We prove the theorem in three steps: 1) a £ d o m / , b £ domdf => [a,b] C d o m / , 2) g r d / C gr<9/, 3) / is convex. 1) Assume that there exists b' £ [a, b] \ dom / ; so b' — Xa + (1 — X)b with A £ ]0,1[. Fix y* £ df(b) and take r £ ffi, r > / ( a ) + A-1(l-A)||6-a||.||2/*||. By Theorem 3.2.5, there exist (xn) - » / c £ [a, b'[ and x* € df(xn) n € N such that liminf (&' — a;n,a;*) > 0,

for every

liminf (b' — a, a;*) > r — f(a).

Since df is monotone we get the contradiction l|6-a||-||y*ll>ll&-c||-||y*||><6-c)i/*> = liminf (b — xn,y*) > liminf (b — xn,x*n) - liminf (A(l - A)" 1 (b' - a,x*n) + (b' - xn, x*n)) > A(l - A) - 1 liminf ( 6 - a , x * ) + liminf (b' >

A ( 1

_

A )

-i

( r

_

/ ( a ) )

-xn,x*n)

.

2) Let x* 6 df{b) with b 6 X. Consider x £ d o m / . Applying again Theorem 3.2.5 for r = f(b), there exist (xn) ->f c € [x, b[ and x* € df(xn) for every n 6 N such that liminf (b-xn,x*n)

> g f f } (/(6) - / ( * ) ) .

Since 9 / is monotone and c = Ax + (1 — A)6 with A € ]0,1], we have that A (6 — x,x*) = (b — c,x*) = liminf (& — x n ,x*) > liminf (b — x n ,a;*) >A(/(6)-/(x)), and so (x — b, x*) < f(x) — f(b) for every x £ d o m / . Therefore x* £ df(x). 3) Let x, y £ dom / and z := (1 — X)x + Xy with A £ ]0,1[. By Corollary 3.2.6, there exists (yn) c dom df such that (yn) ->•/ y. Let a;n := (1-A)x + Ay„ for n £ N; by 1) we have that zn £ d o m / for every n. Using again

184

Some results and applications of convex analysis in normed spaces

Corollary 3.2.6, there exists ((zn,k,zn k)) c STOf s u c r i t n a t (zn,k) —>•/ zn for k —> oo and liminffc_>oo(^n — zn,kiznk) — 0. By 2) we have that z*nk G df(zn>k), and so (x-zntk,z^k)+f(zn,k)

< f(x),

(yn-

zn,k,znk)

+ f{zn>k) < f(yn)

for all n,k G N. Multiplying the first inequality by (1 — A) > 0 and the second one by A > 0 we get (zn - zn,k, z*n>k) + f(zn,k)

< (1 - A)/(x) + Xf(yn)

Vn, k G N.

Taking the limit inferior for k —> oo we get f(zn) < (1 — A)/(x) + Xf(yn) for every n. Passing to the limit inferior for n —> oo and taking into account the lower semicontinuity of / and the fact that (yn) ->/ y we get f(z) < (1 — A)/(a;) + Xf(y), and so / is convex. • We give now another short proof for the Rockafellar theorem on the maximal monotonicity of the subdifferential of a lsc convex function. Theorem 3.2.8 Let X be a Banach space and f € T(X). maximal monotone.

Then df is

Proof. Let x G X and x* ^ df(x). Then x is not a minimum point for g := f — x*, and so there exists z E X such that g{z) < g(x). Let r G E be between these numbers. By Theorem 3.2.5 applied for d on r(X), there exists ((yn,yn)) c Sr9g such that (yn) —>fl y € [z,x[ and lim inf (x - yn,y*n) > | f 5 | [ (r - g{z)) > 0. Therefore (x - yn,y*n) > 0 for n > no, for some n 0 G N. It follows that y* := yno + x* € df(y), with V '•= 2/n0) and (x — y,x* — y*) < 0. Therefore df is maximal monotone. • Theorem 3.2.5 can also be used for proving that two convex functions with the same subdifferential differ by an additive constant. In fact one obtains even more. Theorem 3.2.9

Let C be an open convex subset of the Banach space X jf

and d an abstract subdifferential on T C IK . Let f G T be proper and lower semicontinuous and g G T(X). Assume there exists e G R+ such that df(x) C dg{x) + eUx*

V i G C.

(3.19)

Then C D dom / = C f) dom g and 9{x) - g{y) ~e\\x-

y\\ < f(x) - f(y) < g(x) - g(y) +e\\x~

y\\

(3.20)

Convexity and monotonicity

of

subdifferentials

185

for all x £ C and y £ C fl domg. Proof. Consider 7 > e. By Corollary 3.2.6 we have that d o m / C cl(dom9/), and so C fl domdf ^ 0. Let y £ C fl d o m 9 / and x £ C. Prom our assumption we have that y £ doming, and so y € domg. We are going to prove that /(*) - /(!/) < $(*) - 5(2/) + e II* - 2/11 •

(3-21)

The inequality is obvious if x £ domg or 1 = y, and so assume that x £ domg and x ^ y. Because dg(y) ^ 0, we have that g'(y,u) > - 0 0 for all u £ X. Consider ip : E -»E defined by £ T(E) and is continuous on domy D [0, ||a; — j/||]. It is obvious that '(t) = g'(y + tu,u) for every £ £ domyj. By Theorem 2.1.5(v) we have that lim g'(y + tu,u) = lim
g'(y,u),

and this quantity is finite. Therefore there exists to £ ]0, ||:r — y||[ such that g'(y + t0u,u)
9 g'(x0,u) < 9{V + *>">to - 9{V) + | ( 7 - e ) = g> ~ \f + | ( 7 - e ). — Ipo 2/|| We want to show that

f(x0)-f(y)
(3.22)

Assume that Eq. (3.22) does not hold, i.e.

f(xo) - f(y) > 9{x0) - g(y) + 7 Iko - y\\ • Choose r € E such that r < f(xo) and r - f(y) > g(x0) - g(y) + 7 ||*o - y\\ • By Theorem 3.2.5, there exist (xn) ->f z £ every n £ N such that

\I),XQ\

and x*n £ df(xn)

liminf (a?o -x„,a;*) > — - — — (r - /(j/)).

(3.23) for

186

Some results and applications of convex analysis in normed spaces

Because (xn) —> z we have that (||a;o — x n ||) —> ||xo — z||, and so

]xo-xn\\

I

\\XQ-y\\

Taking into account Eq. (3.23), there exists no G N such that *o-*» - * \ Fo-a;n|| /

>

g(»°)-gy M^cro — 2/11

vn>n0.

+ 7

Since ( x „ ) —>• z G [y, xo[ C C, we m a y a s s u m e t h a t xn G C for n > no- B y E q . (3.19) we have t h a t xn = zn + eu*n with z* G dg(xn) a n d u* G t / x * for n > no- Using also t h e convexity of g, it follows t h a t g(x0)-g(xn) ^ / so^gn ||ar0 — a;„||

\ ||z 0 - z„|l

A

>

9(xo)-9(y)

/

+ 7

_

£

V n

^no_

||ar0 — 3/1

Taking into account the lower semicontinuity of g at z and Eq. (3.19) we get g{xo)-

g(z)

g(x0) - g(y)

,

,

IN-2/H

IFO-^II

Since 2 G [y, io[, z = y + su for some s G [0, io[- Using the convexity of

—ii-


rr- = — :

,

,

— <
(x0,u).

\\x0 - z\\ t0 - s Therefore Eq. (3.22) holds. Consider T~{te

]0, ||rc - y||] I f(y + tu) - f(y) < g(y + tu) - g(y) + jt} .

From Eq. (3.22) we have that t0 G T, and so t := supT G]0, ||a; - y||]. Because

_ , , > * & • « ) = V (<) > V (0) = ff+(»,0),

whence g'(y,u) G 1R. Moreover, y G d o m / D domg. Proceeding as above with y replaced by y (noting that we used only that y G dom / n dom g and g'(y,u) G R), we obtain some s 0 G]0, ||a; - y\\] such that f(y + s0u) - f(y) < g(y + s0u) - g(y) + -ys0.

Convexity and monotonicity

of

subdifferentials

187

As f{y) — f(y) < g(y) — g(y) + 7*, we obtain by addition that f(y +(t + s0)u) - f{y) < g(y + (t + s0)u) - g{y) + 7 ( I + s 0 ), contradicting our choice of t. Hence t = \\x — y\\, and so f(x) — f(y) < g(x) — g(y) + 7 \\x — y\\. Because 7 > e is arbitrary, we obtain that Eq. (3.21) holds. Taking i G C f l dom g and a fixed y G C n dom df, from Eq. (3.21) we obtain that C n dom g c C f l d o m / . Let now j / e C f l d o m / . From Corollary 3.2.6 we get a sequence (yn) C dom df with (yn) —>/ y. Because y G C, we may assume that yn G C for every n € N. Let i G C n dom g be fixed. From Eq. (3.21) we have that

f{x) - f(yn) < g(x) - g(yn) + e\\x-yn\\

Vn € N.

Since g is lsc at y, we get fix) - f(y) < g(x) - g(y) + e\\x-

y\\,

(3.24)

and so y £ dom g. Hence C (~1 dom / = C fl dom g. Let x, y € C fl dom / = C fl dom g; then Eq. (3.24) holds. Interchanging x and 1/ in Eq. (3.24), we obtain also that

f{x) - f{y) > g(x) - g{y) - e \\x - y\\, which proves that Eq. (3.20) holds for all x,y G C f] d o m / . Equation (3.20) being obvious for x & C \ dom / = C\ dom g and y £ Cr\ dom g, the conclusion follows. D A more precise result is the following. Corollary 3.2.10 Let X be a Banach space and d an abstract subdifferential on T C K. . Consider a proper and lower semicontinuous function / € f and g € T(X). If df(x) C dg(x) for every x G X then there exists S;£R such that f = g + k. Proof. Take C — X and e ~ 0 in Theorem 3.2.9. Fixing x0 G d o m / = dom g, we obtain that f(x) = g(x) + f(xo) — g(xo) for every x G X. • Corollary 3.2.11 Let X be a Banach space and f,g€ dg(x) for every x G X then f = g + k for some k €.R.

T(X). Ifdf(x)

C

Proof. Apply the preceding corollary on F(X) and the Fenchel subdifferential. •

188

Some results and applications of convex analysis in normed

spaces

The next result completes Proposition 2.4.3. Corollary 3.2.12 Let X be a Banach space and T : X =4 X* be a maximal cyclically monotone multifunction. Then there exists f G T(X), unique up to an additive constant, such that T = df. Proof. Because T is maximal monotone, g r T ^ 0. Fix (XO,XQ) G g r T and consider fr defined by (2.37) on page 85. Then, by Proposition 2.4.3 we have that fr G T(X) and g r T C grdfr- Since T is maximal monotone we even have g r T = grdfr, and so T = df. The uniqueness follows from Corollary 3.2.11. • 3.3

Some Classes of Functions of a Real Variable and Differentiability of Convex Functions

Consider the following useful classes of functions: A = {

1 + |
A0

Nk = {ip G A | P 3 11-> t~kip(t) € 1 + is nondecreasing}, Jfc G {0,1,2},

= {<^ G yi | iimmrkip(t) = o}, ke {o, i},

r

= {ip G .A |

To Si

= r n A), r^ := y G r 0 1 iiminft_+0Ot-V(*) > o}, = r n n i , s? ~ {^eEi liimsup^^rVC*) <°o}.

Of course, N2 C Ni C N0, E? C Si C fti C ft0 and Tg C T 0 C T C A^. Moreover, if

0 then

(s) := inf{t > 0 | y>(t) > s } ,

0 |
(3.25)

it is obvious that p e < / , i/ie £ iVo, 0} G 1; it is obvious that #(0) = 0 and 0 for s > 0, and so we get a function
Differentiability

of convex

functions

189

Furthermore, one can extend

i(i) := a\t\ with a > 0, *(s) for s > 0, while for s < 0, $5(s) = oo, ipl(s) = t[_ Qi0 ](s), ^ ( s ) = | s 2 ,
(i) Ifip£A0r\N0

then tpe,tph £ Cl0; if(p£Cl0r\N0

then

(ii) Let

i> £ r 0 <s> v # e Si and v e rg o v # e sf • (iv) Let p,q £ ]1, oo[ be such that 1/p + 1/q = 1 and

t~p(f(t) is nondecreasing (nonincreasing) on P then the mapping s i-» s~g R, f(x) := ip(\\x\\). Then f*{x*) = #(||:r*||) for every x* £ X* and f**{x) = <^*#(||a;||) for every x £ X. Proof, (i) Let ip £ A0nN0 and assume that 0 such that ct for every n £ N. So, for n £ N there exists tn > a such that 0 such that a for every n £ N, which contradicts ip £ fio(ii) Consider (for example) the extension ip2 of tp defined above. By what was already noted, we have that 0. Since ip*2* = coip2, we obtain that ip** = coip. (iii) Let

0; then there exists e > 0 such that e. Then ip < ip, and so (p* > ip*- In

190

Some results and applications of convex analysis in normed spaces

particular v*(s) > sup{st - st/2 \t£ [0,e]} - es/2 > 0. Hence 0. For 0 < s < e~lip(e) we have that 0] — max {sup{is - e}} < max{se,sup{i(s — e~1ip(e)) | t > e} — se, and so limS4.0 s~lip#(s) = 0. This means that cpy(t) for all* > 0 and c > 1. Let d > 1 and s > 0. Then dq 0} > sup{(ds)(dq/pt)

- 0} =
Therefore s i-> s~qip#(s) is nonincreasing on P. The proof for the other statement is similar. (v) For x* £ X* we have that r(x*)=sup{(x,x*)-p(\\x\\)\xeX} = sup{(x,a;*) - 0,

\\x\\ = t}

= sup sup ((x,x*) — (p(x)) =sup(t\\x*\\—(p(x)) t>0 \\x\\=t

= tp* (\\x*\\).

t>0

The equality f**(x) = <^ ## (||a;)|| is obtained similarly.

D

As an illustration of the use of the above classes of functions let us give the following interesting result about the differentiability of a convex function. First recall that a bornology on X is a family B of nonempty bounded subsets of X having the properties: 1) every set B of B is symmetric, 2) B is directed upwards, i.e. for all B±, B2 £ B there exists B3 £ B such that S i U B2 C B3 and 3) 1){B | B £ B} = X. The topology induced by the bornology B on X*, denoted by Tg, is the topology of uniform convergence on every set B of B. The most common homologies on X are the Gateaux, the Hadamard and the Frechet homologies which are obtained taking as B the classes of finite symmetric subsets, the weakly

Differentiability

of convex

functions

191

compact symmetric subsets and the bounded symmetric subsets of X, respectively. To the Gateaux bornology corresponds the weak* topology on X*, while to the Prechet bornology corresponds the norm topology on X*. It is clear that the topology m which corresponds to the bornology B is finer than the weak* topology and coarser than the norm topology. The function g : X —>• E is said to be S-diflFerentiable at x if / is finite on a neighborhood of x and there exists x* £ X* such that lim

/(x

t->0

+

^)-/(x) t

\ '

/

'

v

v

the limit being uniform w.r.t. u 6 B, for every B £ B; x* is denoted by Vf(x) and is called the ^-differential or ^-derivative of / at x. Of course, for the Gateaux, Hadamard and Prechet bornologies one obtains the Gateaux, Hadamard and Prechet derivatives, respectively. Condition 3) implies that / is Gateaux differentiable at x whenever / is B-differentiable at x for the bornology B; in particular the ^-derivative is unique when it exists. T h e o r e m 3.3.2 Let f € A(X) be continuous atx£ bornology on X. Consider the following statements:

doxnf and B be a

(i) / is B-differentiable at x, (ii) / is B-smooth

at x, i.e. l i m ^ o i - 1 ^ ^ ) = 0, where, for t > 0,

o-B(t) := sup{/(x + ty) + f{x - ty) - 2f{x) \ y € B}, (iii) every selection of df is norm to T& continuous at x, (iv) for every x* £ df(x) there exists a selection 7 of df which is norm to TB continuous at x and j(x) = x*, (v)

there exists a selection of df which is norm to TB continuous at x,

(vi) (a;*) -4 x* whenever x* £ df(x), (xn) -)• x,

x* £ df(xn)

for n £ N, and

(vii) (x*n) ^4 x* whenever x* £ df(x), and (en) -» 0, (xn) ->• x,

0 < en, x*n £ denf(xn)

for n £ N,

(viii) (x*) ^4 x* whenever x* € df(x), and (e n ) -> 0.

0 < e„, x* £ denf(x)

for n £ N,

TTien (i) <£> (ii) <=> (iii) «• (iv) <s> (v) o (vi) <£= (vii) o (viii). Moreover, if X is a Banach space and f is Isc then (vi) => (vii).

192

Some results and applications of convex analysis in normed spaces

Proof, Then

(i) => (ii) Assume that / is S-differentiable at x. Let u € X.

lim ^ tio

+ tU) +

^

~ ^

~

2f

^

t

= lim f{* + *4.o = (u,Vf(x))

tu)

t +

~

/(5f)

I lim / ( * ~ t\o (-u,Vf(x))=0.

tu)

t

~

/ (

^

As the limits in the right-hand side of the first equality above is uniform w.r.t. u £ B, for every B £ B, we obtain that / is ^-smooth at x. (ii) => (i) Take x* £ df(x); such an element exists by Theorem 2.4.9. Consider B £ B and take u £ B; one has OBH) > f(x + tu)+f(x-tu)

-2f(x)

>f(x

+ tu) - f(x) -t(u,x*)

> 0,

and so 0
tu)-m

_{UiX.}<*BM

vues.

Hence Eq. (3.26) holds, the limit being uniform w.r.t. u £ B. (i) => (iii) Let 7 : D —• X* be a selection of df, where D := dom df; of course, int(dom/) C D and 7(2;) = V/(af). Let B £ B. Consider aB : K+ - > ! + ,

a B ( t ) := sup{f(x + tu) - f(x) -t(u,j(x))

\u£

B}. (3.27) Because / is S-differentiable at x, lim^o £ _ 1 <7B(£) = 0, i.e. a £ fii. Because / is convex, <7j9 is convex, too. For x £ D, u £ B and £ > 0 we have that (z + tu - x, 7(0:)) < /(& + tu) - /(:r) < / ( £ ) + t (u, j(x)) + aB(t) -

f(x),

whence t (u, j(x) - i{x)) - aB(t) < f(x) - f(x) + (x-x,

7(2)).

Taking first the supremum with respect to u £ B, then with respect to t > 0 on the left-hand side of the above inequality we obtain that (CT B ) # (sup u€B

\{U,J(X)

- j(x))\)

< f(x) - S{x) +

< f(x) - f(x) + \\x - x\\ • | | 7 ( i ) | |

(x-x,j(x)) Vx £ D.

Differentiability

of convex

functions

193

Since / is continuous at x, by Theorem 2.4.13, df is bounded on a neighborhood of x, and so 7 is bounded on a neighborhood of x. Prom the above inequality we obtain that l i m ^ ^ ^ ^ s ) * (sup u £ B \(u,7(x) — 7(x))|) = 0. Because, by Lemma 3.3.1, er# G To, we get lim^-yj s u p u e B \(u,f(x) — 7(x))| = 0. Therefore 7 is norm to Tg continuous at x. (iii) =>• (iv) and (iv) => (v) are obvious. (v) =£- (i) Let 7 be a selection of df which is norm to T& continuous at x. Consider B 6 B and e > 0. Then there exists 5 > 0 such that D(x, 6) C D and s u p u € B \(u,j(x) - 7(x))| < e for all x G D(x,6). Because B is bounded, B c /3Ux for some ft > 0. Fix u G 5 and take <£ : K -> 1 , 0. Moreover,
f(x+tu)-f(x)-t(u,j(x))


+ tu,-u)

- (u,7(x)),

< ip'+(t) for every t G int(domy>).

=
+ su)-j(x))

ds
for all t G [0,S']. Hence limt4.oi-1o\B(£) = 0, where WB is denned by Eq. (3.27). It follows that / is S-differentiable at x and V/(x) = 7(1). (iii) => (vi) Let a;* G df(x) and ((x n ,x*)) C gr<9/ be such that (xn) ->• x. Assume that x* is not the limit of the sequence (x*) for the topology TB- It follows that there exist e 0 > 0, B G B and an infinite subset P of N such that s u p u 6 B \{u,x*n — x*)\ > e 0 for every n G P. From (iii) => (i) we have that df(x) = {x*}, and so xn ^ x for n E P. Since (x n ) —> x, we may assume that xn ^ xm for distinct n,m E P- Let 7 be a selection of 9 / such that 7(x n ) := x* for n E P, j(x) := x* and 7(x) an (arbitrary) element of df(x) otherwise. From (iii) we obtain that 7 is norm to TQ continuous at x, and so (x*) n e p TA- x*, contradicting our choice of P. (vi) =>• (iii) Let 7 be a selection of df, and assume that 7 is not norm to rg continuous at x. Then (since x has a countable base of neighborhoods) there exists a sequence (x n ) C D converging to x such that (7(x n )) does not converge to 7(2:) w.r.t. TB; this contradicts (vi). (vii) =>• (vi) is obvious; just take en = 0 for n G N. (vii) => (viii) is obvious; just take for n G N. (viii) =>• (vii) Let x* E df(x), 0 < en and x* G dSnf(xn) be such that

194

Some results and applications of convex analysis in normed spaces

(en) ->• 0 and (xn) -> x. Consider e := sup{e„ | n 6 N}. Since / is continuous at x, by Corollary 2.2.12, / is Lipschitz on a neighborhood of x, while from Theorem 2.4.13 we have that b\f is bounded on a neighborhood of x. Hence there exist r,m > 0 such that \f{x) — f(y)\ < m\\x — y\\ for x,y e D(x,r) and def(D(x,r)) C mUx*- There exists n0 € N such that xn £ D(x,r) for n > n 0 . Let x* € dSnf{x). Then {y - x, x*) = (y - xn, x*) + (xn - x, x*) < f(y) - f(xn)

+en + \\xn - x\\ • \\x"\\

< f(y) - f(x) + m \\x - xn\\ +en + m- \\xn - x\\ for all y € X, and so d

enf(xn)

C den+2m\\xn-x\\f(x)

Vn > n0.

Because (e„ + 2m \\xn — x\\) —> 0, from our hypothesis we obtain that

(OneP^X*Assume now that X is a Banach space. (vi) =^ (vii) Let x* e df(x), 0 < e„, a;* e dSnf(xn) be such that (e n ) -^ 0 and (x„) —> x. Because xn 6 int(dom/) for large n and / and / coincide on int(dom/), we may suppose that / is lsc. By the Borwein theorem, for every n € N there exists (x'n,x'*) e gr<9/ such that \\xn - x'n\\ < ^/E^ and 11a;* - x'*\\ < y/e^. Therefore (a;^) ->• x. By (vi) we obtain that (x'*)n£N -A x*. As (\\xn — x'*\\) -¥ 0 and TB is coarser than the norm topology, we obtain that (a;*)neN TA x*. • Because any selection of df coincides with the gradient of / at the points x where df{x) is a singleton, from the preceding theorem we get the following. Corollary 3.3.3 If f & A(X) is continuous and Gateaux differentiable on the open set D c d o m / then f is B-differentiable on D if and only if V / is norm to TB continuous on D. • As a consequence of Theorems 3.3.2 and 3.1.2 we get the following interesting result. Corollary 3.3.4 Let X be a Banach space and f £ T(X). differentiable atx* 6 int(dom/*) then V/*(aT*) € X.

If f* is Frechet

Proof. By the Br0ndsted-Rockafellar theorem there exists ((a;„,a;*)) C grdf such that « ) ->• x*. Then xn 6 df*(x*n) for the duality (X*,X**).

Well conditioned functions

195

Taking the Prechet homology, from the implication (i) => (vi) of the preceding theorem we get (xn) -» V/*(z*). Because X is closed in X** we obtain that Vf*(x*) G X. • A similar result holds for Gateaux differentiability of /* under supplementary conditions on X. Corollary 3.3.5 Let X be a weakly sequentially complete Banach space and f G T(X). / / / * is Gateaux differentiable at x* G int(dom/*) then V / * 0 T ) G X. Proof. As in the proof of the preceding corollary, there exists a sequence ((xn,x*n)) in grdf such that (a;*) -+ x*. We have again that xn G 9/*(x*) for the duality (X*,X**). Taking now the Gateaux homology, from the implication (i) =>• (vi) of the preceding theorem we get (xn) -» Vf*(x*) for the weak* topology of X**, i.e. ((x*,xn)) -» (x*, V/*(z*)> for all x* G X*. In particular (xn) is a weakly Cauchy sequence, and so, by hypothesis, (xn) converges weakly to some x G X. It follows that V/*(i*) =x € X. • Note that every reflexive Banach space is weakly sequentially complete. The next result shows that for convex functions the usual sufficient condition for Prechet differentiability is also necessary. Corollary 3.3.6 Let f G T(X) be continuous and Gateaux differentiable on a neighborhood of x G d o m / . Then f is Prechet differentiable atx if and only if V / is continuous at x. Proof. We may assume that / is continuous and Gateaux differentiable on the ball B(x,r) C d o m / with r > 0. Because V / can be prolonged to a selection of df, the conclusion follows from Theorem 3.3.2. • Note that for convex functions defined on finite dimensional normed vector spaces Gateaux and Frechet differentiability coincide (use for example Theorem 3.3.2 and take into account the fact that the norm and weak topologies coincide). This is not true for infinite dimensional normed spaces as the functions in Exercises 2.10 and 3.1 show. 3.4

Well Conditioned Functions

Let / G A{X) be such that argmin/ := {x G X \ f{x) = i n f / } ^ 0, •0 G A and 5 C X be a nonempty closed convex set. We say that / is

196

Some results and applications of convex analysis in normed

spaces

^-conditioned with respect to S if MxeX

: f{x)>mff

+ i(>(ds{x))

(see Eq. (3.62) on page 237 for the definition of ds). If there exists rj > 0 such that the above inequality holds for every x € X with ds(x) < T), we say that / is locally ip-conditioned with respect to S. Let us note that if / is V'-conditioned with respect to the subset S and tp £ A0, then argmin/ C S. Taking into account this fact, we say that / is ^-conditioned if / is ^-conditioned with respect to argmin/. Assume that S := argmin/ ^ 0 and consider the conditioning gage of / defined by

V>/:t+ -*•!+,

ipf(t)~mi{f(x)-Mf\xeX,ds(x)=t}.

(3.28)

It is clear that tpf 6 A and / is ^/-conditioned. The following result holds. Proposition 3.4.1 ipf £ Nx. Proof. Let t > exists a sequence ipf(ct). Let (en) I ct+en. Let An :=

Let / £ A(X) be such that S := argmin/ ^ 0. Then

0 and c > 1; we must show that ipj{ct) > cipf(t). There (xn) C X such that ds{xn) = ct and (f(xn) — inf/) —> 0. For every n there exists an £ S such that \\xn — an\\ < (||a;„ — an\\ — ct + t) / \\xn — an\\ 6 [C1,1[. We have that

ds ((1 - Xn)an + A„a;n) > ds(xn)

- (1 - A„) \\xn - an\\ = ct-ct

+ t = t.

Because A i-> ds ((1 — A)an + Xxn) is a continuous function, there exists fin £ ]0, An] such that ds ((1 — £«„)an + nnxn) = t- We obtain so that V'/W < / ( ( I - fJ-n)an + HnXn) - inf / < (1 -fin)f(an)

+ Hnf(xn)

-inf/

= Mn (/(^n) - inf / ) < An (f(x„) - inf / ) . Because (An) -* c _ 1 , we obtain that ipf(t) < c~l • ip/(ct).

O

Let us remark we used only that / ( ( l - X)a + Xx) < (1 — X)f(a) + Xf(x) for every a £ S, x £ X and A £ [0,1], in other words the fact that / is star-shaped at every a 6 5; of course, this condition ensures the convexity of 5. Corollary 3.4.2 Let f and S be as in Proposition 3.4-1- If ip '• [0,a] -> M.+ , where a > 0, is such that Vx£X,

ds(x)
: f(x) > inf / + V

(ds(x)),

Well conditioned functions

197

then f is ipa-conditioned, where ,

™

™

, , ,

[ V>(*) I x^t

for for

te \0,a\, t €ja,oo[.

Proof. From the definition of ipf we have that ipf > ip on [0,a]. For ds(x) = 7 > a we have that / ( i ) > inf / + V/(7) > inf / + -V>/(a) >

inf

/ + ~^(a)

= inf / + ^ d s ( : r ) = inf / + « (d 5 (a:)), a whence the conclusion.

•

Proposition 3.4.1 shows that ipf(t) > 0 for every t > t0 if ipf (to) > 0. It is important the case in which ipf(t) > 0 for every t > 0, i.e. ipf £ Ao', in such a case we say that / is well-conditioned. The following result holds. Theorem 3.4.3 Let X be a Banach space and f S T(X) be such that S := argmin/ ^ 0. T/je following statements are equivalent: (i) £/iere exist ip £ A and to > 0 s«c/t i/rni VW > 0 for every t G ]0, to[ and f is ip-conditioned; (ii) for every sequence (xn) C X with (f(xn)) —> inf/ we have that (ds(xn)) -> 0; (iii) tftere ezisis ip £To such that f is ip-conditioned; (iv) B t r e E i , V x * 6 l * : /*(x*) < /*(0) + tj(i*) + <7(||x*||); (v) 3 V € r 0 , V i e S , V ( z , a : * ) e g r 0 / : ( i - i , a ; * ) > (vi)

B ^ e y i o n A f o , V(x,x*)egrdf

:

ip(ds(x));


(vii) 3flGn 0 n7Vo, V(a;,a:*)egra/ : ds(x) < 9(\\x*\\); (viii) V(z n ) C X : (d 8 / ( T n ) (0)) -> 0 => (d s (x„)) -> 0. Furthermore, in the equivalence (iii) <£> (iv) we can take a = ip&, ip = • (v) we can iafce the same ip, in (v) =£• (vi) we can iafce tp & A, ip(t) = t _ 1 V'(0 /or t > 0, in (vi) =S- (vii) we can take 6 = (vi) we can take ip = 8e and in (vi) =4- (iii) we can iafce ^>(i) = (1 — 7) J0 (p{ls) ds with 7 £]0,1[ fixed. Proof, (i) =£• (iii) It is clear that ipf > ip, and so ipf e A0 (since ipf is nondecreasing). Furthermore, since liminft_HX) t~1tpf(t) > 0 (because

198

Some results and applications of convex analysis in normed spaces

tpf £ A0 fl iVi), we have that ip := cotpf £ T 0 (see Lemma 3.3.1(iii)). Of course, / is ^-conditioned. (iii) => (ii) is obvious, since for ip £ T 0 we have that (tp(tn)) -> 0 =>• (*n) ->• 0. (ii) =>• (i) Suppose that there exists £o > 0 such that V/(*o) = 0- Then there exists (x„) C X such that ds(xn) = t0 and (f(xn) - inf / ) -> 0, which evidently contradicts the hypothesis. As / is ^/-conditioned, the conclusion holds. (iii) =>• (iv) Using Theorems 2.8.10, 2.3.1(ix) and Corollaries 2.4.7, 2.4.16 (see also Exercise 3.7) we have that (V o d s ) * = t £ + ^ # ° | | - | | ,

{t*s + ^o\\.\\)*=^ods.

(3.29)

Using Eq. (3.29), for x* € X* we have that f{x*)

= sup((x,x*)-f(x)) x€X

< sup {(x,x*) - i n f / -

ip(ds(x)))

xeX

= no) +tyo dsy (x*) = no) + L*S(X*)+^*(\\x*\\). The conclusion follows with a := ip#; a € Si by Lemma 3.3.1 (iii). (iv) =>• (iii) is obtained similarly, using Eq. (3.29), with ip := a#. (iii) => (v) Let x £ S and (x,x*) € grdf. Then f(x)>f{x)+ip(ds{x))

and f{x) > f(x) + (x -

x,x*).

Adding these two relations we obtain the conclusion with the same ip. (v) =>• (vi) From (v) we obtain that VxeS,

V(x,x*)

Egrdf

: \\x - x\\ • \\x*\\ > ip(ds(x)),

whence ||x*|| • ds(x) > tp(ds(x)) for every (x,x*) £ grdf. The conclusion follows taking ip € A0 defined by 0. (vi) => (vii) Let 8 := (ph. By Lemma 3.3.l(i) we have that 6 £ N0 n fi0Furthermore, from the definition of 9, we have that #(||a;*||) > ds{x) for every (x,x*) £ Of. (vii) =$• (vi) Let

(iii) Let 7 £]0,1[ and V : K+ -^ W+,

V(*) = ( l - 7 ) * - v ( 7 * ) -

Well conditioned

functions

199

Suppose that (iii) does not hold for this ip. Then there exists x E X such that f(x) < ini f+ip(ds(x)). It follows that x £ S and there exists c E ]0,1[ such that f(x)<mif We take e = c • ip(^j(ds{x))) there exists u E X such that f(u) + e • \\u — x\\ < f(x)

+

c-iP(ds(x)).

> 0. Using Ekeland's variational principle,

and f(u) < f(x) + e • \\u — x\\ Vx E X.

The last relation being equivalent to df(u)C\eUx* i1 0, there exists u* E X* such that (u,u*) E gr<9/ and ||u*|| < e. FVom the first relation, by the choice of x, the definitions of x[> and e, and the hypothesis, it follows that \\u — x\\ < (1 — -y)ds{x)

and
(p(j(ds(x))).

But ds(u) > ds(x) - \\u - x\\ > ds(x) - (1 - j)ds(x)

= 7 • ds(x),

whence the contradiction <^(7 • ds(x)) > c • v?(7 • ds(x)). Hence (iii) holds for this tj). Because t • ^(jt) > JQ (t) := (1 - 7) JQ tp(^s) ds. (vii) => (viii) is obvious. (viii) =>• (vii) Consider the function 6 : 1 + -> ! + ,

6(t) = sup{d s (x) I {x,x*) 6 grdf, ||a;*|| < t}.

It is obvious that 9 is nondecreasing, 6(0) = 0 (i.e. 8 E NQ) and ds(x) < 6(\\x*\\) for every (a:,a;*) E grdf. If limt|o^(*) > 0, there exists £0 > 0 such that 0(t) > £0 for every t > 0. Prom the definition of 6, for every n E N there exists (xn,x^) E df such that ||a;*|| < 1/n and ds(xn) > EQ. Of course this contradicts the hypothesis. D We remark that we used the convexity of %[> only for the implication (iv) => (iii); for the other implications it is sufficient that ij) E N\. The function ipf verifies this condition. Also, we used that X is a Banach space only in the implication (vi) =>• (iii). We also note that in the implications (iii) <$ (iv), (v) =>• (vi) <S> (vii) •£> (viii), (vi) =*> (iii) =>• (ii) =*> (i) we can

200

Some results and applications of convex analysis in normed spaces

take S C X a nonempty closed convex set, and in the implication (iii) => (v) we used only the fact that S C argmin/. Corollary 3.4.4 Let X be a Banach space, f € T(X) and (x, x*) € grdf. The following statements are equivalent: (i) 3iPeT0,

Vx€X

: f(x)>f(x)

(ii) 3 f f € S i , V i * 6 P

: f*(x*)

(iii) 3 ^ € r 0 , \/(x,x*)Ggrdf (iv) 3ip£A0nN0,

(x-x,x*)+rP(\\x-x\\);

< f(x*)+{x,x*

- x*)+a(\\x* - x*\\);

: (x - aT,x* - z*) > V(||a; - z||);

\/(x,x*)egrdf

(v) 3den0nN0,v(x,x*)egTdf (vi) V(xn)cX

+

: y>(||x - z | | ) < ||x* - x*\\;

•.

\\x-x\\ <eq\x* -x*\\y,

: (da/(:Cn)(x*))->0=^(:En)->z;

(vii) x* 6 int(dom/*) and / * is Frechet differentiable at x*. Proof. Taking the auxiliary function g : X —> R denned by g(x) = f(x) — (x, x*), we may assume that x* = 0. The equivalence of conditions (i)-(vi) follows from Theorem 3.4.3 by taking S := {x} (taking into account, eventually, the remarks done after the proof of that theorem). (ii) =>• (vii) Taking into account that (x, x*) £ gr<9/, we have that Vi'eT

: 0 < f*(x*)

- f*(x*)

- (x,x* -x*)


whence x* £ int(dom/*) and lim sup

r(x*)-r(x*)-(x,x*-x*)

< l i

m s u p

t|0

t

^ = l i m ^ = 0 . *4-° t

This shows that / * is Frechet differentiable at a;* and V/*(a;*) = x. (vii) => (ii) Because (x*,x) € g r ( 9 / ) _ 1 C gr<9/* (the last one for the duality (X*,X**)), we obtain that x e df*{x*). Since / * is Frechet differentiable at x*, we have that V/*(aT") = x. For t € K+ consider a(t) := suv{r{x*)

- f*{x*) - (x,x* -x*)

= sup{r(x*+tu*)-r(x*)-t(x,u*)

\ \\x* -x*\\ = t} | | K | | = 1}.

Because the function 11-> f*(x* + tu*) - f*(x*) — t {x,u*) (6 M.+) is convex and lsc, it follows that a G T. Furthermore, from the Frechet differentiability of the function / * it follows that lim^o £-1
Well conditioned

functions

201

A sufficient condition for (i) from the preceding corollary is that / ( ( l - X)x + Xx) + A(l - X)rp(\\x - i||) < (1 - X)f(x) + Xf(x)

(3.30)

for all x £ X and A £ ]0,1[. Indeed, if Eq. (3.30) holds and x € dom / then {f{x + X(x - x)) - /0c))/A < f{x) - f{x) - (1 - X)^{\\x - x\\) for all x £ X and all A 6 ]0,1[, whence Vxel

: f{x)>f{x)

+ f'{x,x-x)+ip{\\x-x\\),

(3.31)

and so VieX,Vi*ea/(i)

: f{x)>f{x)

+ (x-x,x*)+iP{\\x-x\\).

(3.32)

A function / e A(X) for which there exists ip &Ao satisfying condition (3.30) is said to be uniformly convex at x (6 d o m / ) . Let / 6 A(X) and x € d o m / . The gage of uniform convexity of / at x is related to relation (3.30), and is defined by Pf,x{t) '•= i n f \ —

A)/(z) + Xfjx) - / ( ( l - X)x + Ax) A£]0,1[, A(l - A) \\x — x\\ = t

= inf { <* ' X»W

+

Y(f_+^ ~ ^

+ A

«»> | u e SX, A 6 ]0,1[} (3.33)

(with the convention oo - oo = oo). Using Theorem 2.1.5(vii) one obtains easily that pf^ € No (Exercise!). We may introduce also a gage related to relation (3.31). More exactly consider tif,x(t) ••= inf {f(x) - f(x) - f'(x, x - x) | x E dom / , ||a; - x\\ = t} = inf {f(x + tu) — f(x) — tf'(x, u) | u e Sx H cone(dom / - x)} . Because the mapping M+ 3 11-> f(x+tu) — f(x)—tf'(x, u) 6 M+ belongs to T C N\, we obtain that j ? ^ € Nx. As in the proof of (3.30) =* (3.31) we have that i9/,j > pf^. Thus, when / is uniformly convex at x € d o m / (i.e. P/,x 6 ^o) then $/,j € .Ao- One may ask if the converse implication holds. The answer is affirmative when / is Frechet differentiable at x € dom / .

202

Some results and applications of convex analysis in normed spaces

Proposition 3.4.5 Let f 6 F(X) be Frechet differentiable at x 6 d o m / . If flf,x € AQ then f is uniformly convex at T. Proof.

Let t > 0 be such that D(x, t) C int(dom/). By hypothesis

f{x)>f{x)

+ f'{x,x-x)+6

VxeX,

\\x-x\\ = t/2,

where 6 = i9/ iS (i/2) > 0. On the other hand, because / is Prechet differentiable at x, a selection 7 : dom df -> X* of df exists which is continuous at x (by Theorem 3.3.2). Hence there exists n > 0 such that Il7(*) - V/(3c)|| < S/t for ||z - x\\ < j]. Let a e]0,1[ be such that 1 - a < 2t}/t. Consider x 6 X arbitrary satisfying ||a; — x\\ — t and take z := ax + (l-a)^^= x-\-±=^-{x-x) € int(dom/). Hence \\z -x\\ = ^^-t < r], and so \\j(z) - Vf(x)\\ < 5/t. Of course, f(x) > f(z) + f'(z,xAs |(37 + x) = -^hx + -^tiz,

z) = f(z) + / ' (z, we

l

-^(x-

x)) .

(3.34)

have that

STT/(z) + S T l / W > / ( § ( * + *)) •

(3-35)

Multiplying Eq. (3.35) with a+1 and adding the result to (3.34) we obtain f(x) + af(x) > (1 + a ) / (I(3f + x)) + f (z, ±=2(3f - x)) , whence f{x) + f{x) - 2 / {\{x + x))>^

(/ (1(3- + a:)) - f(x) + \f< (z,x - x)) 6

>^(

+ y^x-x)

+ I / ' (z,3f- a;))

>^(<5+i) ^^(^-IIIV/^-TWII) > ^ • I = ^ 0 > 0. Therefore f(x) + f{x)-2f{\{x + x))> 80 for every x E X with ||a; — a;|| = t. The conclusion follows like in Exercise 2.24(iv). • Without Frechet differentiability of / at x the result in the preceding proposition may be false (see Exercise 3.4).

Uniformly convex and uniformly

3.5

smooth convex

203

functions

Uniformly Convex and Uniformly Smooth Convex Functions

We "globalize" the notion of uniformly convex function at a point as follows. We say that the proper function / : X —> K is p-convex, where p £ A, if / ( ( l - X)x + Xy) + A(l - X)p(\\x - i/H) < (1 - X)f(x) + Xf(y). for all x, y £ X and all A £ ]0,1 [. Of course, if / is p-convex then / is convex; we say that / is uniformly convex if there exists p £ AQ such that / is p-convex. We say that / is strongly convex (with rate c or c- strongly convex) if / is pc-convex for some c > 0, where pc £ AQ, pc{t) •= \ct2. It is clear that when / is uniformly convex then / is uniformly convex at every point of its domain and when / is uniformly convex at every point of its domain then / is strictly convex. When / £ A(X) and A C X is a convex set we say that / is uniformly convex on A if / + LA is uniformly convex. It is natural to introduce the gage of uniform convexity of the convex function / . Namely, it is the function p/ : K+ -»• R+ defined by p/(i)=inf|

(l-X)f(x)+Xf(y)~f((l-X)x

+ Xy)

A(l - A)

AG]0,1[,

x,y £ d o m / , \\x — y\\ = t > . Of course, Pf > p when / is p-convex. Hence / is uniformly convex if, and only if, p/ £ Ao. The gage of uniform convexity of the convex function / has a remarkable property. Proposition 3.5.1

Let f £ A(X).

Vt>0, Vc>l i.e. the function 11-» t~2pf(t)

Then :

Pf(ct)>c2pf(t),

is nondecreasing on P.

Proof. Let t > 0 and c £ ]1,2[ be such that p/(ct) < oo. Let us fix e > 0. There exist x,y £ d o m / and A G]0, | ] such that ||y — x\\ = ct and n , ^ Pf(ct)

. . (l-X)f(x) + pe >

+ Xf(y)-f((l-X)x ^ — -

+ Xy)

.

204

Some results and applications of convex analysis in normed spaces

Consider x\ := (1 - X)x + Xy and xc := (1 - c l)x + c ly. We have that \\xc — x\\ — t and x\ = (1 — c\)x + cXxc (of course cX 6 ]0,1[). Hence f(xc)

< (1 - c-1)^)

f(xx)

< (1 - cX)f(x) + cXf{xc) - cA(l -

f(xx)

> (1 - X)f(x) + Xf(y) - A(l - X)pf(ct) - eX(l - A).

+ c-'fiy)

- c-\l

-

c-^pfict), cX)pf(t),

From these relations (multiplying the first with cX > 0) we obtain that c2Pf(t) < Pf(ct) + ec• ±Eh < PM)

+£-£~C-

Letting e -> 0, we obtain that c2/9/(£) < pf(ct) for all c € ]1, 2[ and t > 0. By mathematical induction one obtains immediately that c2npf(t) < Pf(cnt) for c e ] l , 2[, n £ N and t > 0, and so c2pf(t) < pf(ct) for every c > 1 and t > 0. The conclusion holds. D A dual notion (as we shall see) for uniform convexity is that of uniform smoothness. Let / € A(X), a £ A and O ^ i c d o m / . We say that / is er-smooth on A if for all x,y £ X and all A 6]0,1[ such that (1 — A)a; + Xy £ A we have / ( ( l - X)x + Xy) + A(l - X)a(\\x - y\\) > (1 - A)/(i) + A/(y),

(3.36)

or equivalently

f(x) + A(l - AMIMI) > (1 - X)f(x - Xy) + Xf(x + (1 - X)y) for all z £ A, y £ X and A £]0,1[; we say that / is cr-smooth if / is cr-smooth on dom / . Having in view this definition, it is natural to consider the following gage of uniform s m o o t h n e s s o n A of the function / : af,A{t)

••= s u p ^

(1 - A)/(:r) + Xf(y) - / ( ( l - X)x + Xy) A€]0,1[, A(l - A) x,y£X,

(l-X)x

+ Xy£A,

f (1 - X)f(x - Xty) + Xf(x + (1 - X)ty) - f(x) sup

\\x-y\\=t\ Ae]Q>1[j

A(l - A)

x £ A, y £ Sx

}•

(3.37)

Uniformly convex and uniformly

smooth convex functions

205

c/.dom / is denoted by CTJ and is called the gage of uniform smoothness of / . It is obvious that 07,4 < a if / is cr-smooth on A. Sometimes it is useful to consider the following midpoint gage of smoothness: a°fA(t) := sup {f(x) + f{y) - 2f{\x + \y) \ \x + \y £ A, \\x - y\\ = 2t} = sup {f{x - ty) + f{x + ty) - 2f{x) \ x £ A, y £ SX}-

(3.38)

It is easy to obtain that 2cr°fA{t/2) < af,A(t) < o-°fA{t) for t > 0. As above, c^dom/ w m ^ e denoted simply 0. / / o~f,Aito) < °° then A + toBx C d o m / ; if a f {to) < 00 then d o m / = X and o-f £ l \ Proof. Suppose first that 07(*o) < 00 and take x € A. From Eq. (3.37) we obtain that x — Xtoy £ d o m / for every A £]0,1[ and y £ Sx, and so B{x,to) C d o m / . Therefore A + toBx C d o m / . If A = d o m / we obtain that d o m / + nt0Bx C d o m / for every n £ N, whence d o m / = X. In this case the functions £ i-» /(a; — Xty) and £ >-» f(x + (1 — A)fa/) are convex and finite, and so continuous; from Eq. (3.37) we obtain that oy £ I \ • The relationships between uniform convexity and uniform smoothness are given in the following result. Proposition 3.5.3 and p,cr £ A.

Let f £ A(X), h £ A{X*) have proper conjugates

(i) J / / and h are p-convex then f* and h* are p# -smooth, respectively. (ii) / / / and h are cr-smooth then f* and h* are o# -convex, respectively. Proof, (i) Suppose that / is p-convex. Let t > 0 be fixed and consider x* eX*,y* £ Sx* a n d A £ ] 0 , l [ ; t a k e ^ := x*-\ty*, x\ := x* + (l-X)ty*. Consider also 70 < f*{xo) and 7J < f*{x{). From the definition of / * there exist Xo, xi £ X such that 70 < {X0,XQ) - f(x0),

71 < (xi,xl)

-

f{x{).

Multiplying the first relation by 1 — A and the second one by A, then adding them to both sides of the (Young-Fenchel) inequality 0 < f(xx) +f*{x*)

~{xx,x*),

206

Some results and applications of convex analysis in normed spaces

where x\ :— (1 — X)xo + Xx\, we obtain that (1 - A)7o + A71 < f(xx) + /*(**) - (1 - A)/(ar0) - A / f o ) + A(l - A)* (x, < f*(x*)

+ A(l - A) (t \\x0 - i i | | - p(\\x0 - nil))


+

x0,y*)

\(l-\)P*(t).

Letting 7; —>• f*(x*) for i = 1,2, we obtain that /* is p # -smooth. The same calculation shows that h* : X —> E is /9*-smooth if /i is p -convex. (ii) Suppose now that / is cr-smooth. Consider x^,x\ 6 X* and A 6 ]0,1[. For x, y € X, denoting x*x := (1 - X)XQ + Az*, we have that

>(l-X)((x-Xy,x*0)-f(x-Xy)) +

X({x + (l-X)y,x*1)-f(x

> (1 - A) (x - Xy,X*0) + X(x+(1-

+

(l-X)y))

X)y,x\) - f(x) - A(l - A)a(||y||)

= {x, x*x) - f{x) + A(l - A) ((y, x\ - x*0) -

a(\\y\\)).

Since x, y € X are arbitrary, we have that

(i-x)r(x*0) + xr(x*1) > snpxeX

((x, x*x) - f{x)) + A(l - A)supJ/€js: ((y, x{ - XQ) -
= rix*x) +

Xil-X)o-#i\\x*-x*1\\),

which means that / * is cr#-convex. The proof for the fact that h* is cr#-convex when h is cr-smooth is similar. • Corollary 3.5.4 Let f 6 T(X). Then o{* = ipf)* and 07 = ( p / . ) # Furthermore t !-• t~2afit) is nonincreasing on P. In particular, if there exists to > 0 such that 07 (£0) < 00 then a fit) < 00 for every t > 0. Proof. From the preceding proposition we have that /* is (p/) # -smooth # and (<7/) -convex, while / = /** is (p/.) # -smooth and (07.)* -convex. Therefore we have that 07 < (/?/•)* a n d Pf > (07)*. Passing to the conjugates in the second inequality we obtain that 07 < (p/*) # < (07)**. Therefore erf = (p/») # = (07)** (in particular we have obtained again

Uniformly convex and uniformly

smooth convex functions

207

that aj is convex and lower semicontinuous). In a similar way we obtain that 07. = (/9/)*. Taking into account Proposition 3.5.1, Lemma 3.3.1(iv) and the relation Of = (p/*)*, we have that t >-> t~2a;{t) is nonincreasing on P. The remainder of the conclusion is immediate. • We say that / G A(X) is uniformly smooth (on the nonempty subset A of dom / ) if there exists a G fii such that / is cr-smooth (on A). When A is a singleton {x} C dom / , / is uniformly smooth on {x} if and only if / is smooth at x in the sense of Theorem 3.3.2. Taking into account Proposition 3.5.2, if / is uniformly smooth o n ^ C l then there exists £o > 0 such that A + SQBX C d o m / , while if / is uniformly smooth then d o m / = X. Summarizing Propositions 3.5.1-3.5.3 (and using Lemma 3.3.1(iii)) we obtain the following important result. Theorem 3.5.5

Let f £T(X).

Then

(i) / is uniformly convex 44> / * is uniformly smooth; (ii) / is uniformly smooth •$=>• f* is uniformly convex. It is quite natural to say that the proper function / : X —> E is uniformly Frechet differentiable on 0 ^ A C X if there exists £o > 0 such that A + SQBX C d o m / , / is Frechet differentiable at any point of A and nm /Or+

t-H)

*»)-/(*) t

V M

uniformly with respect to x € A and u € Sx- Of course, when A = {a} this means that / is Frechet differentiable at a. Similarly to the proof of Theorem 3.3.2, one obtains easily that / G A(X) is uniformly Frechet differentiable on A C dom / if and only if / is continuous at some point of its domain and is uniformly smooth on A. The following result gives other characterizations of uniformly smooth convex functions. Theorem 3.5.6

Let f G A(X). Consider the following

statements:

(i) / is uniformly smooth; (ii)

( d o m / ) ! / 0 and there exists a<x G fii such that

V x G ( d o m / r , V y G X : f(y) < f{x) + f{x,y-x)

+ a2{\\y - x\\); (3.39)

(iii) d o m / = X, f'(x, •) is linear for every x G X and there exists &$ €

•

208

Some results and applications of convex analysis in normed spaces

Oi such that \/x,y£X

+ f,(y,y-x)
: f'(x,x-y)

(iv) grdf ^ 0 and there existsCT4G fti

SMCA

(3.40)

i/ioi

y(x,x*)egrdf,

V j / e X : /(i/)+a4(||y-a:||); (3.41) (v) dom 9 / = X and there exists as G fii such that V(x,x'),(v,v')£&df

• (y-x,y*-x*)
(3.42)

(vi) dom df = X and there exists ipi G To such that

v{x,x*),(v,v')e&df

: m>f(x)

+ (y-x,x*) + M\\y*-x'\\); (3.43)

(vii) dom df = X and there exists V>2 € T 0 suc/i i/iai V (*,**), (y,y*)egidf

: (i/-i,y*-a;*) > ^(||y*-i'||);

(3.44)

(viii) dom df — X and there exists y> £ Ao(l No such that \/(x,x*),(y,y*)

£grdf

: \\y - x\\ > V{\\y* - x*\\);

(3.45)

(ix) dom df = X and there exists 6 £ flo f) No such that V(z,z*), ( \\y* - x*\\; (x) d o m / = X, f is Frechet differentiate $7I such that Vx,yeX

: (x-y,

Vf(x)

on X and there exists &§ G

- V/(j/)) < a6(\\y - x\\);

(xi) d o m / = X, f is Frechet differentiate uous.

(3.46)

(3.47)

and V / is uniformly contin-

Then conditions (i) - (iii) and (iv) - (xi) are equivalent, respectively, and (iv) =£• (ii). Furthermore, if f is continuous at some point of dom / (or X is Banach space and f is lower semicontinuous) then the eleven statements are equivalent; in such a case if one of conditions (i) - (xi) holds then dom / = X and f is continuous. Proof, (i) =>• (ii) Let a G fii satisfy Eq. (3.36); we already observed above that d o m / = X. From Eq. (3.36) we have that A" 1 (f{x + \{y - x)) - f(x)) + (1 - AMHi, - x\\) > f(y) -

f(x)

Uniformly convex and uniformly

smooth convex functions

209

for all x,y G X and A G]0,1[; taking the limit for A -¥ 0 we obtain Eq. (3.39) with cr2 : = a(ii) => (iii) Let x0 G ( d o m / ) \ From Theorem 2.1.13 we have that f'(xo,u) G R for every u G X. Let t0 > 0 be such that <72(£0) < oo. From Eq. (3.39), it follows that {y G X \ \\y - x0\\ = t0} C d o m / , and so B(x0,t0) C d o m / (because d o m / is a convex set). Therefore (dom/) 1 + toBx C dom / , whence, as in the proof of Proposition 3.5.2 we obtain that d o m / = X. Changing x and y in Eq. (3.39), then adding side by side the corresponding relations, we obtain

V i j e l

: 0
+ f'(y,x-y)+2a2(\\y-x\\).

(3.48)

Let x G X be fixed and u £ X \ {0}. Consider the function