Theory of Vector Optimization

¢wc(z) y) = sup{z(x.) + y(b) - ¢J(x, b) : ; and the conjugate dual is of the form: (D*) sup -¢»; (0, y) S.t6 Y E Yl.

E E 17 b E

E3 }

134

where :Yl is the space of continuous linear maps from E a to E, that is according to the notations used in Section 1.

Yl

~

C

CQrollary 4.9 (Azimov, 1982) If X o EEl, Yo E Yl are solutions of (P) and (D*) and if the optimal values of these problems are equal, then X o and Yo are connected by the re lation

¢J(x o , 0) + *(0, Yo) =

o.

Conversely, if X o E E 1 and Yo E Yl satisfy the above relation, then (P) and Yo solves (D*), and the op-timal values are equal.

(4.4) $0

solves

Proof. In order to apply Theorem 4.8, set A = {(x, O) : x EEl}, B = {-4>*(0, y) : y E JIl}. Since 4>* is the conjugate map of 4>, relations (1 (x)) and (2( x)) cannot simultaneously be true. Therefore, if X o E X and Yo E Yi satisfy the relation (4~4), then (AR') holds. The result is now drawn from Theorem 4.8~ •

Chapter 6

Structure of Optimal Solution Sets

In mathematical programming if an optimization problern has an optimal solution, then the optimal value of the problem is a unique point in the field of real numbers. Due to this fact, the set of optimal solutions has rather simple structure whenever the data of the problem are convex or linear. For instance, the optimal solution set of a linear problem is a polyhedron, the optimal solution set of a convex problem is convex if of course it is noncmpty. Such a ~icc structure of solution sets is no longer true for vee.tor problems. However, some properties '\vhich are important in application such as closcdness, connectedness or contractibility of these sets are still in hand if the data satisfy certain assumptions. This chapter is devoted to the study of the properties mentioned above of efficient point sets for linear, convex and quasiconvex vector problems.

1.GENERAL CASE

Let E be a real topological vector space and C a convex cone in E. We consider the following vector problem: minf(x)

s.t.x EX,

136

where X is a nonempty set equiped with a topology and f is a point-valued map from X to E~ We recall that S(X; f) denotes the set of optimal solutions and W S(X; f) denotes the set of weakly optimal solutions of the problem above. Theorem 1.1 Assume that the interior of C is nonempty~ If X is closed, C-continuous on X J then W8(X; f) is closed.

f

is

Suppose to the contrary that there is a net of weakly optimal solutions X o which is not weakly optimal. Then, since X is closed, ~o E X, there is some x E X such that .

Proof.

{x a } converging to

f(x o ) - f(x) E intC. Let V be a neighborhood of f(x o ) in E such that V ~ f(x) + intC. By the C- continuity of !, there is an index 0'0 such that j(xaJ E V + C, for all 0:' > 0'0' Comhine this with (1.1) to get the C9n tradiction: X a ¢ WS(X;!), completing the proof. -

(1~1)

Corollary 1.2 If f(X) is closed, then WMin(f(X)fC) is closed~ In particular, if X is compact, f is continuous, then W1\din(j(X)IC) is closed~ Proof. If I(X) is closed, then according to Theorem 1.1, the set WS(!(JY); id) is closed, where id is the identity map on J(X) . Hence ltV1\.1in(f(X)JC) is closed because it coincides with the set WS(f(X); id).

Finally if X is compact,

f

is continuous, then f(X) is closed.•

It should be worthwhile to note here that the results above are not always true for the sets SeX; f) and 1.VIin(f(X)JC). lfthe space E has dimension greater than one, then there are compact sets with the efficient point sets being noncompact. Take for example in R2 the set consisting of the unit ball {(x,y) E R 2 : x 2 +y2 $1} and the point (-2,0). Then the efficient point set ,vith respect to the cone R~ is the union

{(-2,O)} U{(x,y) E R2: x 2 +y2 = l,x S O,y and it is not compact.

< O}

137

2.LINEAR CASE

Let A be a polyhedron in Rn, C a polyhedral cone in Rn. We simply usc Nlin(A) instead of Min(AIC) in thc sequel by keeping in rnind that the order is . generated by the cone C~ We also recall that a subset A o ~ A is a face of A (in a general sense) if either A o = A or there exists a hyperplane H in the space such that A is entirely contained in one of the half spaces gcnerated by II and such

that A o = AnH.

.

Definition 2.1 A set B ~ Rn is said to be pathwise connected if for every a, b E B, there are a finite number of points in B: bo = a, b1 , ••• ) bl + 1 = b such that the segments [b i ) bi+l], i = 0, ... , l belong to B.

Theorem 2.2 For a given polyhedron A in Rn, the sets l\1Iin(A) and W Min(A) if they are nonempty, consist of certain faces of A and they are pathwise connected~ Proof. It is known that for cvery linear function ~ on R n , the set of points minimizing ~ on A is a closed face of A. Hence the first assertion of the theorem is deduced from Theorem 3.3 (Chapter 4)~ The second assertion can be obtaind from the first one and from the connectedness results of the next section concerning convex sets. However we shall also furnish an alternative proof by using the argument of Poclinovski-:\'ogin(1983). Let A 1 , ••. , Ak be relatively opcn faces of A) not meeting cach other and their union is Min(A)~ Take ai E Ai and consider the closed convex cones C*(ai) = {~ E Rn : ai E S(A;~)}) i = 1) ~ .. , k.

If denote

co =

{~ E ri(C*) : min{~(x)

:x

E A} exists },

then

co ~ G*(al) U ... U C*(ak).

(2.1)

Indeed, for ~ E Co, the solution set S(A;~) is a closed face of A and it belongs to lWin(A). Hcnce,there is some i such that

Ai ~ S(A;E). This means that ~ E C:ic(ai)~ Xow let a) b E l\!Iin(A) with a E Ai, b E A j , some i, j E {I, ... ) k}. We have to show that there are some bo , ~ •. ) bl+ 1 E 1vJ in(A) such

that (2.2)

138

[b r , br +1] ~ 1\1in(A), r = 0, ... , L

In fact, by Proposition 3.2 (Chapter 4), there are some ~a E Co n C*(ai),~b E Co n C*(aj) such that a E S(A; €a), b E S(A; ~b).

It is clear that Co is a convex cone, hence

[€a, ~b] S;; Co. In view of (2.1), there arc some ~l = {a,€l

=

€1, ... ,~l E Co such that

~b,

[{a, ~bJ = [~l, ~2] U ... U [~l-l' ~d and [~r, ~r+l] ~ C* (ai(r)), where i(r) E {I, HO' k}, r = 1, ,.. )1- 1.

(2.3)

It foIlo,vs from (2.3) that ~T+l E G*(ai(r))

n C*(ai(r+l») nCo,

and therefore [ai(r), ai(r+l)] ~

SeA; ~r+l) ~ j\1in(A). (2.4) Since c;l = ~a, a E S(A; €1); and since a E Ai, Ai ~ S(A; c;l). In other words, ai{l) = ai and (2.5) [a, aiel)] ~ S(A; c;l) ~ Min(A). Similarly, [ai{l) ,

b] ~ S(A; C;l) ~ i\1in(A).

(2.6)

Now, setting bo = a, br = ai(r), bl+ 1 = b, r = 1, ... , I and using relations (2.4), (2.5) and (2.6) we obtain (2.2).

For WMin(A) the proof is analogue.• Let us now consider a linear problem

minf(x) s.t.x EX, where X is a polyhedron, f is a linear function from X to Rn. If S(X; f) (resp., WS(X; f) ) is nonempty, then it consists of certain closed faces of X and it is pathwise connected. .

Theorem 2.3

ProoE. This theorem is proven by a similar way as the preceding

on~.•.

139

3.CONVEX CASE

Let A be a convex set in a topological space with an ordering cone C. The aim of this section is to study the sets Min(AJC) and WMin(AIC). First we recall some defmitions from topology~

Definition 3~1 Let A ~ B ~ E. We say that A is 1) contrac ti bIe if there exist a continuous junction H (a, t) from the produc t A x [0, 1] to A and a point a o E A such that

H(a.,O)=a o , H(a,l)=a/orallaEA; 2) locally contractible provided any neighborhhod V of any point a E A contains. a neighborhood U of a such that UnA is contractible; 3) a retract of B if there is a continuous function h from B to A such that h( a) = a for each a E A ~ L~mma

cone ~ap

3.2 Suppose that E is a locally convex space, C is a closed convex pointed nonempty interior and A is a closed convex set. Then the set-valued G from A + C to E defined by the rule: w~th

G(a) = (a - C) n (A + C), fOT each a E A

is continuous at every point of int(A; + C) U Min(A1C).

+ C,

Proof. The upper continuity of G is obvious because it is the intersection of a continuous map a H a - C and a constant map a H- A + C. To prove the lower continuity, let first a o E Min(A1C), if the latter set is nonempty.Then

G{a o ) = {a o }. Hence, for any neighborhood V of a o , by taking U = V we see that

G(a) n V

f 0, for each a

E U,

whic.h means that G is lower continuous at that point. Now let ao E int{A+C) and suppose to the contrary that the map is not lower . continuous at that point, i.e there are a neighborhood V of some point bo E G(a o ), a .net {aa} from A + C converging to a o such that

G(a Q )

nV

=

0.

(3.1)

·Since ao E int(A + C), there is a convex neighborhood, say W of ao such that

W

~

int(A + C).

140

Let W o be the convex hull of bo and W4 'fhis set belongs to A + C since bo E A + C and the set A + C is convex. Y.[oreover,

bt = tb o + (1 - t)a o E (intWo ) n V,

(3.2)

\vhcncvcr t is smaller than and close to I. Let U be a neighborhood of bt in the right hand side set of (362). Then a o E U + C and there is some index a o such

that aa

Let

Ua:

E U,

Since U

~

Co:

Wo

E U + C) for all a 2;:

Qo·

(343)

+ C, for aU 0:' ~ 0:'0.

(3.4)

E C' be such that

~

aa: = U Q + Ca A + C, '\ve have that aa: E A

It follows from (3.3) that U ce

= aa -

CO'

E ao: - C,

which together with (344) gives the inclusion Uo:

E G( a cr ), for a C 0'0.

'I'his contradicts (3.1) and completes the proo[ •

Remark 3.3 It should be observed that under the conditions of Lemma 3.2, the + C , but it is not necessarily lower continuous there.. To see this, let us consider the following example in R3 . The cone C = -R~, the se t A is the con vex hull of the point (0, 1, 1) and the set {(x,y,z) E R3 : y = o,x 2: O,z;::: O,x2 +z2 :51}.

map G is upper continuous on A

The map G is not lower continuous at the point (0,0, 1). This example also shows that the efficient point set of a convex compact set in R3 is not necessarily compact_ We recall that a function 9 from E to R is strictly quasiconvex if it is strictly R+-quasiconvex (Definition 6.1, Chapter 1), Le. for every t E R, x" Y E E and x f y, g(x) ~ t, g(y) :5 t imply g((x + y)/2) < t4 In other words, 9 is strictly quasiconvex if its level set at any p oint of R is strictly convex when being nonempty. Lemma 3.4 Let E be a normed space with a strictly quasiconvex norm and C a convex cone with convex bounded base. 'l'hen one can construct a continuous function 9 from E to R which is strictly {j'uasiconvez and increasing with respect to (C, R+).

141

"Proof. Let B be a convex bounded base of C . By a separation theorem therc is vector ~ E intC' such that

", a

~(b)

> c, for all b E B,

some c

> O.

Denote

H{t) = {x E E : ~(x) = t}; B o = {b/~(b) : b E B}. "Then, B o S; If(l), and it can bc verified that B o is also a convex bounded base of C . Let k be a positive number such th~t Eo ~ intB(O; k), 'where B(O; k) is the ball in E with the center at 0 and radius k. Consider now the set

(3.5)

1) = {x E E : c;(x) = t, IIxli :5 k(t + t 1 / 2 ), t ;?: O}. "Ve claim that D possesses the following properties:

(x + y)/2 E intD, for each x, y E D, x # y; C \ {O} ~ intD.

(3:6)

(3.7) To establish the latter relation, let e E C,c f= o. Then ~(c) = t > 0 and by (3.5), C E intB(O; kt)~ One can then find a positive number () such that x E intB(O, kt), fo! all x E E, [Ixl1 < b. (3.8) Let 'Y = min{6; «(1

+ 2t)1/2 -

1)/2}

and set

M = {x E E : ~(x) > t -" IIx - ell < ,}. It is obvious that I is positive and M is an open set containing c. Moreover, M ~ D. Indeed, since t > 0, for any x E M,

{(x) > t -

r

~

t - ((1 + 2t)1/2 - 1) > 0,

and by (3.8),

tlxll

~ kt S k(t - 1) + (t -1)1/2)

< k(~(x) + (c;(X))1/2). These two relations say that xED and hence (3.7) is proven. Before proving (3.6), let us state a useful observation, which can be obtained by a way similar to that we have uscd in proving (3.7) :

x E intD if there is some t E R such that

(3.9)

€(x) = t > 0 and Ilxll < k(t + t 1 / 2 ), As for the relation (3.6), let x, y E D, x;f y and c;(x) = t, ~(y) = s. Since x -;;f. y, at lea..~t one of the nonnegative numbers t and 5 must be positive. If t = s, then

142

c;((x + y)/2) = t

>0

and by the assumption on the norm,

II(x + y)/211 < k(t + t 1/ 2 ). This and relation (3.9) show that

(x + y)/2 E intD. 1= 5, then e(x + y)/2) = (t+s)/2 and we can estimate the norm of (x + y)/2 as !I(x + y)/211 5 Irxll/2 + l[yll/2 :5 k(t + 5 + t 1 / 2 + 8 1 / 2 )/2 < k((t + 8}/2 + «(t + 5)/2)1/2). Again, in vie\v of (3.9), (x + y)/2 E intD, establishing (3~6). Further, if t

With D in hand we are able to proceed to construct g by the rule: g(x) = inf{t E R : x E te - D}, x E E, where e is a fixed vector from B o • We claim that 9 is well defined. Indeed, the set {t E R : x E te - D} is nonempty, for instance any positive number being greater

thaD. max{~(x);

€(x)

+ (~(x) + IIxU/k)2}-

belongs to it. Moreover, that set is bounded from below because any number being smaller than ~(x) does not belong to it. In this way, g(x) is correctly defined.

All that remains to be proved is that g is continuous, increasing and strictly . quasiconve:"(. We establish first the following fact: for every x E E, t E R,

x E int(te - D) if and only if 9(x) < t. Indeed, if x E int(te - D), then there is some K > 0 such that x + Ke E int(te - D).

(3~10)

Consequently,

x E (t - ~)e - D, implying g(x) ~ t - "'~ < t, then, by (3.7) we have that x E (t - (t - g(x))/2)e - D ~ te - ((t - g(x))/2)e - D ~ te - intD ~ int(te - D). Now, the increasingncss of 9 is derived from relations (3.7) and (3.10); its strict Conversely, if g(x)

quasiconvexity is derived from relations (3.6) and (3.10). As for the continuity, let

{x ce } be a net in E converging to X o E E. It is clear that {g(x a )} is bounded. Let t be a cluster point of this net. In virtue of (3.10), t 5 g(x o ). If t > g(x o ), then, again by (3.10), we arrive at the contradiction:

g(x o ) < t - (t - g(x o ))/2, for a large enough.

143

Thus, limg(x a )

= g(x o ), completing the proof. •

- Concerning the assumption on the norm in the lemma above, in the literature a strictly quasiconvex norm is called strict normJt is known that in Banach separable spaces there always exist strict norms. More generally, if in a locally convex space there is a strictly convex bounded absorbing neighborhood~thenthe Minkowski functional defined by that neighborhood provides a strict norm. Let A be a compact set in E with A + C being convex. Then under the assumptions of Lemma 9.4, the function 9 attains its minimum on A at a unique poin t.

Corollary 3.5

Proof. Since A is compact and 9 is continuous, min{g(x) : x E A} exists, say it is t. Let x,y E A with .

g(x) = g(y) = t. If x ;f y,. by the strict quasiconvexity, g(x

Further, since A

+C (x

+ y)/2) < t.

is convex,

+ y)/2 E A + C.

,lIenee there is some z E A, c E C such that (x + y)/2 = z + c. Remembering that .9 is increasing we arrive at a contradiction:

g(z)

~

g((x + y)/2) < t.

The proof is complete. •

Theorem 3.6 i) E is a ii) C is a iii) A is a

Suppose that the following conditions holds: space with a strict quasiconvex norm; convex cone with nonempty interior and convex bounded base; convex set with Ax = (x - C) n A being compact for each x E E~

Then lv[in (A IC) is contractibIe. Proof. First let us construct a function h from A

+C

to M in(AJC) as fo110\\'5:

h(x) = {y E G(x) : g(y) = min{g(z) : z E G(x)}}, for x E A

.

(3.11)

+ C. Observe that 9 is increasing, therefore we can have the equality: min{g(z) = z E G(x)} = min{g(z) : z E Ax}.

The value in the right hand side of the equality above is finite because 9 is continuous and A z is compact. By Corollary 3.6, the set in the right hand side of (3.11) is a single point which in view of Proposition 1.2 (Chapter 4) belongs to Min(AfC)~

144

Thus~ h is correctly defined. Further, by Lemma 3.2 and by the continuity of 9, the function h is continuous on int(A C) U Min(AIC). Moreover,

+

G(x) = {x} for each x E Min(AIC)' hence

hex) = x.

(3.12)

We are now in a position to construct the map H from l\1Iin(AIC) x [0,1] to Min(AIC). Let Us fix a point a E int(A+C) and for each x E Min(AIC), t E fO, 1] "\ve set

lI(x, t) = h(tx + (1 - t)a). It is clear that II is continuous in both variables~ Moreover, by (3.12), H(x, 1) = h(x) = x, and H(x,O) = h(a) E Min(ArC), for all x E l\1Iin(ArC). In this way, Min(AIC) is a contractible set and the proof is complete.• Theorem 3.7 Under the conditions of Theorem 3.6, lvlin(AJC) is closed if and only if it is a retract of A + C . Proof. Since A + C is closed, any retract of this set is closed. Kow, suppose that l\1in(AIC) is closed. Denote by d(x) the distance from x to the set l\1Iin(AIC) and let

t(x) = d(x)/(l Construct a retract~on

+ d(x).

f from A + C to i\lfin(AIC) by the rule

. f(x) = h(l - t(x))x + t(x)a), where h is the function defmed in the proof of 'I"\heorem 3~6, a is a fixed point from int(A + C). For any x E A + C \ve have (1 - t(x))x + t(x)a E int(A + C) U l\IIin(AIC). Therefore, f continuous on A + C . Moreover, if x E Min(ArC), then f(x) = hex) = x. Consequently f is actually a retraction and the theorem is proven.•

is

As a matter of fact, the result of Theorem 3~6 can be improved when the space is of dimension 2, as the following theorem shows. Theorem 3.8 Assume that A is a compact convex set in R2. Then Min(AJC) is homeomorphic to a simplex. Proof. To prove this, we argue first that for every nondegenerated affine transformation 1~ of R 2 ,

145

T(lv.fin(AJC)) = Min(TA[TC) .

. In fact, it is clear that T is increasing with respect to (C, TC) and the inverse transformation T-l is increasing with respect to (TC, C). The equality is now dra\vn from Proposition 1.2 (Chapter 4)~ Further, ,ve consider the first case where the aimension of C is equal to two and C is closed. By the observation just made above it can be assumed that C = R~ and A ~ intC. For the closedncss of Min(A]C), let {ak} be a sequence from that set converging to a EA. If this point were not efficient) then there would be some b E A with a >c b, say a l > b1 , a2 ~ b2 , where the upper indexes . indicate the components of the point~ Since ak arc efficient and lim ak = a, a% > b2 when k is large enough. For a fixed large number n,

[an,b] ~ A+C and a E [an, b] + intC. This would imply that ak cannot be efficient whenever k is large. The contradiction establishes the closcdness of Min(A1C). Next, we show that the cone generated by this set is convex. Indeed, suppose that a and b are two efficient points and c = ta + (1 - t)b, 0 ~ t ~ 1. Consider the optimization problem

mint s.t. t 2:: 0, tc E A

+ C. It is clear that this problem has an optimal solution, say to. We claim that toe is efficient. Indeed, if not, there is a point x E A such that toe> x. Observe that the triangle with vertices a, b and x is contained in A + C and it c~ntains toe as an interior point. This fact contradicts the choise of to. l~hus, the cone generated by Min(ArC) is convex closed. Moreover it is pointed since C is the nonnegative orthant. Consequently, it has a base homeomorphic to a simplex~ .Direct verification shows that the base is homeomorphic to Min(AIC) and therefore the latter set is homeomorphic to that simplex too. For the case \vhcre C is not closed one can consider C as the cone R~ \vithout one or both extremal rays. The proof described above goes through without change~

For the case where the dimension of C is one, there is no loss of generality if we suppose that C' is the ~lIst coordinate axis and as before,A ~ intR~. Let Xo =

min{x 2 : x =

(X l

,x2 ) E A+C, xl = O},

Yo = max{ x 2 : x E A + C, Xl = O}.

It is clear that these solutions exist and they are unique and one can verify without any difficulties that Min(AIC) is homeomorphic to the segment [x o, Yo]. The proof is complete. •

146

Remark 3.9 The result of Theorem 3.8 can fail when the dimension of A is higher than two ~ as is ShO\VD by the follo\ving example in R 3 . Let A be the polyhedron with the vertices: (3) 2, 0), (2,3,0), (4,0,0), (0,4) 0) and (2.6, 2~6, 3).

It i.s a convex compact set and its efficient point set with respect to the nonpositive orthant consists of t \Vo triangles, one of which is with vcrtices

(4,0,0),(3,2,0),(2.6,2.6,3) and another is with vertices (0,4,0), (2,3,0), (2.6,2.6,3).

'I1hese triangles have only the point (2.6, 2.6,3) in common and their union cannot be homeomorphic to a simplex. We return now to the set WMin(AIC). Recall that a set X is arcwisc connected if for any two points x, y E X, there is a continuous function t.p from [O,lJ to X such that ~(O)

= x,

cp(l) = y. Theorem 3.10 connected.

Under the conditions of Theorem 3.6, WMin(AJC) is arcwise

Proof. Let a b E WMin(ArC). It is clear that there are x,y E Min(A'C) such 1

that a ~c x and b 2.c y.

Moreover, [a,x] and [b,y] belong to WMin(ArC). According to Theorern 3~6, the set Min(AIC) is contractible, hence arcwisc connected. Let ep be an arc linking x and y in lVlin(ArC). Then the arc

[a, x] U

Corollary 3.11 Suppose that E = Rn, C is pointed closed with nonempty interior and A is closed convex. Then WJ\l.lin(AIC) is arcwise connected if Min(A'C) is nonempty.

Proof. If Min(ArC) is nonempty, then condition iii) of Theorem 3.6 holds. Other conditions are satisfied trivially. Kow invoke the corollary to Theorem 3.10.•

147

Remark 3.12 In the preceding corollary if lVIin(AIC) is empty, then the set WMin(AIC) may be disconnected. Indeed, let us consider the following example

rn R 3 • Let C

be the nonpositive orthant, A the convex hull of the points:

(O,O,O),(O,l,O),(l,O,O),(O,O,k),(O,l,k),(l,O,k) and (1-1/2 k ,1-1/2 k ,k), k=1,2, ... ~ Then

1\11 in(AiC) == 0,

WMin(A1C) = {(l,O:z): z ~ O} U {(O,l,z): z ~ O}.

Theorem 3a13 Under the conditions of Theorem 3.6, WMin(AIC) is locally contractib Ie if and only -if it is a re tract of A. Proof. Since A is a convex closed set, it is locally contractible, and so is any retract of A. Now·, suppose that WMin(A1G) is locally contractible. By Dugundji (1958, Theorem 15.1) it follows that WMin(AlC) is a retract of some neighborhood V of WMin(A1C) in A, Le. there exists a continuous map hi from V to WMin(AlC) as in Definition 3.1. Further, for every positive integer k there exists a number s(k) > 0 such that

h2 (x) = (x + sd(x)h(x))J(l + sd(x)) E V, for each x E A with lixll :5 k, s ;::: s(k), where h is the function constructed in Theorem 3.6, d(x) is the distance from x to W.l\.1in(A1C). Without loss of generality we may assume that

s(k + 1)

> s(k) for

every k.

Let

s(t) = (s(k + 2) - ·s(k + l))t + (k + l)s(k + 1) - ks(k + 2)~ for t, k :S t :5 k + 1, k = 1", 2, ... , and let h 3 (x) = (x + s(l1 x lDd(x)h(x))/(l + s(l1xIDd(x)). It is obvious that h 3 is a continuous function on A~ Moreover, h 3 (x) E V for all x E A and h3 (x) = x whenever x E WMin(AIC). Now the composition h 1 0 h3 will be the continuous function on A to W M in(AIC) as required in Definition 3.1. In this way WMin(A1C) is a retract of A .• We finish this section by a short comment on the conditions of Theorem 3.6. Conditions i) and iii) are of no specification. However, condition ii) is rather strict. For instance if E is a separated Ricsz space (i.e. for every two finite sets A, B ~ E with the property: a ~ b for all a E A, b E B, there exists some x E E such that a ~ x ~ b, for all a E A, b E B), then condition ii) implies that the space is finite dimensional (sec Jameson-1970).

148

4.QUASICONVEX CASE

In this section let us consider a quasiconvex problem:

min/ex) S.t.x EX, where X is a convex closed set, f is a C-quasiconvex function from X to E , and G is a convex pointed closed cone in E with nonernpty interior. Throughout this section ,ve shall make use of strictly increasing continuous functions from E to R which were introduced in Section 4, Chapter 1 for every a E E, e E intC = he,a(Y) = min{t : yEa + te - C}, y E E. V\'e shall sometimes write h instead of he,a if it is clear that a and e are given. Lemma 4.1 If in addition f is C-continuous, then for every fixed points a E E, e E intC~ the solution set B( X; he,a (.) 0 f) is closed convex" Proof. Note first that the scalar function ho f is quasiconvex in view of Proposition 6.3 (Chapter 1). Moreover, by Theorem 5.11 (Chapter 1), it is upper semicontinuous. The result now is immediate.•

Lemma 4.2 If in addition to the assumption of Lemma 4.1, f is strictly Cquasiconvex, then the set S(X; h 0 f) is a single point whenever it is nonempty4 Proof. In this case the function h immediate.•

0

f

is strictly quasiconvex and the lemma is

Lemma 4.3 Let a be a fixed point of E and let lev(a) denote the level set of f at a. If f is C-quasiconvex and the weak solution set WS(lev(a); f) is compact, then the set-valued map

e H S(X; he,a(.) 0 f), e E E is upper continuous on intC. Proof. Suppose to the contrary that the map above is not upper continuous at a point eo E intC. This means that there are: a net {eo:} from intC converging to eo, a net {xa}, XI): E S(X; heCHa.(.) 0 f) and a neighborhood U of SeX; heo,a(') 0 f) in X such that X Q ¢ U. It can be seen that

S(X; he~ta(.) 0 f) ~ WS(lev(a); f).

149

-Therefore, without loss of generality we may assume that {xQ-} converges to some E X~ Moreover, f is C-continuous, by Corollary 5.1 (Chapter 1),

~ ~o

Xo

. Since

XC(

E lev(a)~

¢ U, we conclud.c that X

o rt SeX; hco,a(~)

which means that there is some

It is obvious that

Xl

0

f)

E X such that

to = hco,a(!(X o)) > he~,a(f(Xl)) = t l . E lev(a). Now it follows from (4.1) that !(Xl) E toe - C,

(4.1)

Xl

. or equivalently,

f(x o ) ¢ tle o - C. Hence there are a neighborhood V1 of f(x o ) and V2 of tle o in E such that Vl

n (V2 -

C) = 0,

also

(Vi + C) n (V2 - C) = 0. Further, since f is C-continuous and limeo: ~ eo, f(x a ) E V + C and t1eo- E V, whenever By definition of h, heCt,a(f(x~))

for

Q

Q'

is large enough.

> heO;,a(f(Xl)),

being large enough. This contradicts the fact that

E S(X; hee-,a(') . completing the proo[ • XC(

0

f),

Before going further, recall that a set A are no open sets U and V in E such that

~

E is said to be connected if there

A ~ U u V, An U # 0, A n V =F 0 and A nun V = 0. Lemma 4.4 Suppose that Ak

~

Ak+l and Ak' k = 1,2, ... are connected sets in

E ;. Then the union U {Ak : k = 1, 2, .. ~ } is connected.

Proof. This is immediate from the definition of connectedness.•

Lemma 4.5 Let F is an upper continuous set-valued map from a connected set D to E with F(x) being connected for each xED. Then F(D) is connected.

150

Proof. Direct verification completes the proof, or see Warburton (1983), Hiri~t­ Urruty(19S5) .•

Theorem 4~6 Assume that f is C-continuous, C-quasicon1Jex on a convex closed set X and WS(lev(a); f) is nonempty compact for every a E E with lev(a) being nonempty. Then WS(X;f) is a nonempty closed connected set.

Proof. That ltV S(X; f) is nonempty is guaranteed by the fact that

WS(lev(a); f) ~ W8(X; f), which is a direct consequence of Proposition 2.8 (Chapter 2). rrhc closedness·of that set is obtained by Theorem 1.1. 1\O\V we have to prove the connectedness. Let us fi."'C a E intC and set ak = ka~ k = 1,2, and ~ct Xk = lev(ak). It is clear that

u.

U{Xk : k = 1, 2, ... }; T¥S(Xk; f) S; WS(Xk+l; f) and WS(X; f) = U{WS(Xk; f) = k = 1, 2, ... }~ In virtue of Lemma 4.4, it suffices to prove that WS(X k ; f) is connected. v~ic fix k and consider a. set-valued map F from intC to WS(X k ; f) as follows: X =

F(e) == S(Xk ; he,ak+l 0 f). Since f(Xk) ~ int(ak+l - C), by Theorem 2.16 (Chapter 4) and Proposition 3.2 (Chapter 4), F(intC) = WS(X k ; f). Now apply Lemmas 4.1, 4.3, 4.5 and use the connectedness of intC to derive the connectedness of WS(Xk; f) .•

vVc recall that for a E E, f(X)a is the section of f(X) at a, te. f(X)a = f(X) n (a + C)~ Lemma 4.7 IjWA1in(f(X)aJC) is compact, then the

set~Jt}alued map

e H S(j(X)a; he,a(.)), e E intC, is upper continuous on intC. Proof. Since he,a(Y) is strictly increasing in y, we have that

S(f(X)a; hc)a(.») ~ WMin(f(X)a'C) which implies

151

We

a1r~ady

know that h is a function being continuous in all variables, therefore

:by Theorem 4.5 (Chapter 4), S(f(X)a; h) is upper continuous in e on intC.• Theorem 4.8 Assume that f is continuous C-quasiconvex on a closed convex set X and WMin(j(X)aI C ) is nonempty compact for each a E E with f(X)a being nonempty. Then WMin(f(X)lC) is a nonempty closed connected set. Proof. 1.'\hat the set W Min(f(X)IC) is closed and nonempty is obvious. For the connec.tedness, the proof is similar to that of Theorem 4.6 by using Lemma 4.7 iristead of Lemma 4.2, a sct--valued map:

. G(e) = S(f(Xk); he ,ak+l (.)) instead of FCe) and by observing that G(e) is connected as the image f(F(e)) of the connected set F( e) under the continuous function f .• Remark 4.9 It should be emphasiied that the two conditions: .i) W M in(f(X)a Ie) is compact; . ii) W S(lev(a); f) is compact, not derived. from each other, neit"her arc Theorems 4.6 and 4.8. To sec this let us consider the following examples: 1) Let

are ..

and lc~

f :X

1--4

X = {(Xl)X2) E R2: 0 ~ Xl::; 1,0 ~ X2:5 I}, R 2 be defined by f((Xl' X2)) = (Xl, 0) if Xl =f 1, X2 =f:. 1; J((l, 1)) = (0,2) .

.Obviously, for C = -R~ rrheorem 4.6 holds meanwhile f(Wlltin(X; f)) = f(X) is disconnected.

2) Let

Let

f

X. bci the convex hull of the sets:

{(x)O)O) E R3 : x ~ O}, {(x,D, 1) E R 3 : x ~ O}, {(x, 1,0) E R 3 : x ~ O}, {(x, sin 2- k 1r, cos2-k1f) E R3 : x !:: k}) k = 1,2, .... be the projection 'from X to R2 as follo,vs:

f((x, y, z) = (y, z) and G = -R~. In this case, ~heorem 4.8 holds, ho'\vever, WS(X; f) = B U (U{Ak : k = 1,2, ... }),

where

~52

x ~ k, (y, z) E [sin 2- k 1f, cos 2- k Jr]}, B = {(x, y, z) E R3 : x ~ 0, y = 0, z = I},

Ak = {(x, y, z) E R3

:

i.e. ltVS(X; f) is disconnected. The first example also shows that without continuity of f the result of Theo·rem 4.8 may fail.

If Wlvlin(f(X)aJC) is nonempty compact for every a E E with j(X) being nonempty and f is C-continuous strictly C-quasiconve::c on X , then both of the maps defined in Lemmas 4.3, 4.1 are point·-vaiued,continuous on intC.

Lemma 4.10

Proof. Due to Lemma 4.2, the maps mentioned above arc point-valued; Hence they are continuous whenever being upper continuous as set-valued maps. The proof of their upper continuity is straightforward, so we omit it.• Assume that f is C-contin'uous strictly C~·quasicon'Vex on a convex closed set X and one of the following conditions holds: i) S(lev(a); f) is compact for each a E E; ii) l\1Iin(f(X)afC) is compact for each a E E.

Theorem 4.11

Then S(X; f) is nonempty closed connected. If in addition f is continuous then Min(f(X)JC) is nonempty closed connected. .

Proof. The proof is similar to that of Theorem 4.6 by using Lemma 4.10 instead of Lemmas 4.3 and 4.7.•

Theorem 4.12 Suppose that f is continuous strictly C-quasicon'Vex on a convex closed set X and M in(f(X)a Ie) is compact for each a E E. Then S(X; f) (resp" Min(f(X)IC)) is a retract of X (resp., f(X»). Proof. The aim is to construct a retraction 9 from X to S(X; f) . Let e be a fIXed vector from intC and let x E X. Consider the problem P(x) :

min he,!(:r:) (f(y») s.t. Y E X. In view of Lemma 4~2, this problem has a unique solution which is denoted by x•. Set g(x) = XofC. We prove that 9 is continuous on X . Indeed, let {xa:} be a net from X converging to X o E X~ Since f is continuous,

limf(x(:l) = f(x o ). :NIoreover,{ he ,f(:to-) (I (x Q))} converges to he ,/(:c 0) (I (x 0») sinee h is continuous in all the variables. If g(x o )

tJ. S(X; he,!(x c )(·) 0 f),

153

.~then

there is some x E X such that he,!(x o ) (f(x))

.By the continuity of h anc:l

> he,!(x o ) (f(x o)) .

!,

he,!(:Co)(f(x)) :> he~f(:r:Of)(f(xo.)), whenever a is large enough. This contradicts the fact that g(x a ) belongs to the . set S(X; he,!ex a ) ( . ) 0 f)· Thus, 9 is continuous on X . The relation g(x) = x for all x E S(X; f) is obvious and 9 is indeed a retraction. A retraction from f(X) to Min(j(X)IC) can similarly be constructed9 •

Theorem 4.13 Under the aS8umptions of Theorem 4.12, Min(j(X)IC) and S(X; f) are homeomorphic to each other~ Proof. We prove that the restriction of f on S(X; f) provides a homeomorphism. between two sets above. For, let y E l\-fin(f(X)IC), there is some x E S(X; f) so that f(x) = y. This x is unique because if there were some z E SeX; f), z ~ x .with fez) = y, then

+ z)/2)

E f(x) - intC, contradicting x E S(X; f). Set g(y) = x to get a function from Min(f(X)IC) to S (X; f) . It can be verified direct that 9 is continuous on Min (/ (X) IC) and g 0 f as well as fog axe identity functions on S(X; f) and Min(f(X)IC), respectively. The proof is complete. •

f«(x

. Corollary 4.14 Under the assumptions of Theorem 4.12, the sets Min(f(X)IC) and S(X; f) are contractible.

ProoE Invoke this corollary to Theorems 4.12, 4.13 and the fact that a set is alwaJo:s contracti hIe. •

convc~

Remark 4.15 If f is continuous C-quasiconvex on a compact convex set, then it is not n~cessary for SeX; f) and Min(j(X)IC) to be connected. For instance, let

x = [-1, 1}, let

f :X

H

R2 be defined by the rule J(t) = (0, -t) if t ~ 0; . f (t) = (t, 0) if t sO;

Then Min(f(X)IR~) = {( -1,0) U

S(X; f) = {-I} U{I}.

(0, -I)},

154

As the conclusion of this section the following remark on the efficient point sets of fractional vector problems should be usefuL Let us consider the following linear fr actional problem: min((alx

+ b1)/(CIX + d~), .... , (anx + bn)j(cnx + dn ))

Sit. X EX, ,vhere X is a polyhedron in Rm, ai, Ci E Rm, bi , d i E R and the ordering cone in Rn is the nonpositive orthant. Since the problem above is quasiconvex, all the results ,vc have established concerning solution sets of quasi convex problems are also vaJid for this problem. Chao and Atkins (1983) succeeded in proving a stronger result. Namely, if X ·is compact; then WS(X; f) is pathwise connected, while the set S(X; f) is connected in the case n S 3. For higher dimension the question is still open.

Comments Chapter 1:

The first six sections are based on Dedieu (1978), Luc (1987 d~e,f), Pomcrol (1985), Precupanu (1984), Yu (1974). Section 7. Most definitions concerning set-valued maps are taken from Aubin and Ekcland (1984), Berge (1963), Penot (1984). Chapter 2:

Section 1. An excelent book on binary relations and preference orders in multiobjcctivc decision making is Yu (1985).See also Percssini (1967). Section 2. The results of this section are classical except for Proposition 2.9 (Lac (1988.a)). For several concepts of proper efficiency see Sawaragi et aL (1985). Section 3~ The general theorems (Theorems 3.3, 3.4) on the existence of efficient points are new (Luc (1988c)). Other results have been obtained by Do!"\vein (1983), Corley (1980), Henig (1982b), Jahn (1986), Sterna- Karwat (1986a,1987c). Section 4. The first work dealing with the domination property is Vogel (1977). Further investigations have been achieved by Benson (1983), Henig (1986), .Luc (1984,1988a). Section 5. Vector problems with set-valued objectives and constraint have .been studied by several authors when developing duality theory. The form we present in this section was given in Corley (1987), Luc (1988c). Chapter 3:

Contingent cones have been introduced by Bouligand (1930). Contingent derivatives have. been developed in Aubin (1981), Aubin and Ekeland (1984), :·Frankowska (1985), Penot (1984)) Ward and Borwein (1987) and others. .. The concept of semidiffercntiabilities and the results of the first three sections .have been given in Luc (1988b)w Necessary and sufficient conditions of differentiable and ( or) convex vector problems have been obtained by many authors,see for example Aubin (1979),Kuhn and Tucker (1951), Da Cunha and Polak (1967), Luc (1985a), Smale (1974), Zalincscu (1987) and others~

156

Chapter 4: The first three sections contain the infinite dimensional version of the results from Luc (1986b,1987b), which include also results of Pascoletti and Serafini (1984), Warburton (1983), Wierzbicki (1977). Some results of Jahn (1984) are also added. For the separation of nonconvex sets see also Gertewitz and Ivanov (1985), Elster and Gorfcrt (1986). Stability has been investigated by Naccache (1979) ,Penot and Sterna- Karwat (1986), Stcrna···Karwat (1987a), Tanino and Sawaragi (1980b)~

Chapter 5: Duality of linear vector problems which is not included in these notes has been considered in Gale et al.(1951), Iserman (1977,1978), I
Chapter 6: The structure of optimal solution sets of linear problems has been obtained very early (Arrow ct al.-1953, Chernikov-1968). The structure of efficient point sets of convex sets has been studied by Bitran and :YJ:agnanti (1979), Luc (1985b), MOIOZOV (1977), Peleg (1972). In Section 3, we furnish the results from the works mentioned above in infinite dimensional spaces. For quasiconvex problems the results have been obtained by Choo and Atkins (1983), Luc (1987c), Warburton (1983). The differentiable structure of efficient point sets is not included in these notes. It can be found in Luc (1983), Schecter (1978), Smale (1976).

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Index acute cone: 2 algebraic dual space: 7 algebraic polar cone: 7 alternative: 57 arcwisc connected set: 146 asymmetric binary relation: 37 asymptotic cone: 9 axiomatic duality: 120 base: 4-7 binary relation: 37,38 boundedly order complete space: 47,48 C-boundcd set: 13,14,15 C·-·closed map: 33~34 C-closed set: 13-17,34 C-compact set: 13,14,34 C-complete set: 46,4.7,58 C--continuous function: 22-24, 27 C-continuous set-valued map: 33 C-convex function: 29,30,32 C-convex set: 15 C-quasiconvex function: 29-32 C-scmicompact set: 14,34 C-scmicontinuQus function~ 22 Cherbyshev norm: 21 closed function: 22 closed convex bounded base: 4-6

compact map: 35,36 complete binaxy relation: 38 complete lattice: 133 complete scalarization: 95,96 composition of functions: 27.,28 composition of set-valued maps:

36 condition (CB): 10~·13,16 condition (CD): 10-13 cone: 1 cone continuous function: 22 cone convex function: 29 cone monotonic function: 18 conjugate dual: 117,133 conjugate map: 117,133 connected binary relation: 38 connected set: 149 constrained problem: 70 constraint : 57 constraint qualification: 75,76 contingent cone: 63 contingent derivative: 63 continuous map: 33--35 contractible set: 139,143,153 convex hull: 1 convex problem: 61,91 convex weakly compact base: 7 correct cone: 2·-4 Daniell cone: 47 Decreasing net: 46

172

Dini lower derivative: 64 Dini upper derivative: 64 direc ti onal derivative: 69. domination property: 53-55 dual map: 110 dual pair of solutions: 112 dual problem: 111,121 duality relation: 129,133 efficient point: 39-44,46 efficient solution: 57 epi-closed function: 22,25-27 epigraph: 18 exact dual: 121,123-125 existence of efficient points: 46···52 extreme vector: 8 feasible solution: 59 feasible triple:· 70,112 feasible COllpIe: 112 fr actiollal problem:· 154 generalized Slater condition: 75 graph: 33 greatest lower bound; 132

ideal efficient point: 39~·43 increasing function: 18-21,81 Lagrangean duality: 110 Lagrangcan map: 111 lattice: 132 least upper bound: 132 level-closed function: 22,25,27 level set: 18 lexicographic cone; 2,39 linear binary relation: 38 linear problern: 61,88~138 linear representation: 88,89 Lipschitz map: 35

local efficient point: 40 local minimizer: 68-73 local optimal solution: 69 local weak minimizer: 68,71 locally contractible set: 139,147 lower bound: 132 lower C-continuous map: 33 lower cquiscmicontinuous family; 22,23 lower scmicontinuous function:

22,24-26 lower scmidiffcrcntiablc map:

62--65 minimal solution: 57 Minkowski functional: 21,93 monotonic function: 18

necessary condition: 71 nondecreasing function: 18-21 nondominatcd point: 39 nonnegative orthant: 2 nontransitive binary relation: 37 "N"-valued ma.p: 33 optimal solution: 57 optimal value: 57,60 Pareto minimal point: 39 partial order: 38 parthwise connected set: 137,153 perturbation: 117,133 pointed cone: 2,24 polyhedral cone: 4,24 positive linear functional: 20 positive linear operator: 20 primal problem; 111 properly efficient point: 39,44,90 properly increasing function: 81 properly optimal triple: 112,113

173 '-

quasiconvex function: 29,30 quasiconvcx problem: 61,92,93,148 quasiconvex representation: 92,93 recession cone: 8-13 reflexive binary relation: 37 relation of the alternative: 129 retract: 139~144,147

saddle point: 115,116 saddle point theorem: 115 scalar proper representation: 87 scalar strict representation: 86 scalar ,veak representation: 87 section: 43 self·-cfficicnt map: 111 separation: 81-85 set·~valued rnap: 33 Slater condition: 111 smallest strictly monotonic function: 21,30 stable problem: 119 strictly C-convcx function: 29,32 strictly C--quasiconvex function; 29 strictly convex problem: 61 strictly increasing function:

19-21

strictly quasiconvex function: 61 strictly supported cone: 2 strongly C--complete set: 46,47 strong duality: 113 subdifferentiable map: 119 sufficient condition: 67,71,77 symmetric binary r~lation: 37 topological dual space: 7 topological polar cone: 7 transitive binary relation: 37 ubiquitous cone: 2)39 unconstrained problem: 67 upper bound: 132 upper C-·continuous map:

33 upper semidifferentiable map: 63-66 vector optimization problem: 57

weak domination property: 56 weak duality: 112,118 weak duality a"'Ciom: 121 weakly efficient point : 40~ -.44 weak separation: 82

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Vol. 237: Misspec·t:cation Ara:ysis. Proceedings. 19B3. EdIted by T. K. Diji<stra. V. 129 pages. 1984.

~ol. 212~ R.

Vol. 238: W. DOr:1$chko. A. Drexl. Locat~ofl and layout P~ann:ng. IV, 134 pages. 1985.

Sato and T. I\~no. Invariance Structure o~ Tecr.nology. V, 94 pages. 1963.

P~i:lc;p'es

ar.c

t~e

Vol. 213: Aspira~:or Levels in Bargai nil1g and EconoT.;c Decision Making. Proceedings. 1982. Edi~ed by R. Tietz. VIII. 406 pases. 1983.

'vo~. 214:

M. Faber, H. Nicr:1es und G. Stepl'-an. F.r.trop:o, Umwelt-

SC~1l;1Z und Rohstctfvcrbrauc"l. IX. 18~ Seiten_ 1983.

Vor. 215: Ser:1i·lnf:n:~e P':ogramr:1ing a.nd App::cat:ons. Proceedi!1gs.

1981. Edited by A. V. Fiacco and 1<.0. Kortar.ek. XI. 322 pages. ~983.

:11 a General EqJilibrilJm Model 'w:~h por$is~e.'t U:'B'Tlp"oyrr.ent. VI. 92 pages. 1983.

.Vo!. 216; H. H. MUlier, Fiscal Policies

Voi. 2 17: C~. Grootaert. Tho Relation Between Final Den-.and aroc !r.come Distribution. XIV, 106 pages. lQ83. Vol. 21 B: P. van Loor:, A Dynamic Theory of the Firm: Prcdt.:ction. Financo ard Inves~mont. VII. 191 pages. 1983. Vol. 219: E. van Darnnc, Ref:r:.e:r.~ntsof tho Nash Equi;ibr:~m Concept. VI, 15: pages. 1983.

Vol. 239: Microeco:":omic Mode s of housi:;g Marke~s. Eo ted K. Stahl. VII. 197 pages. 1985.

by

Vol. 240: Contribufons to Opera.t:ens Rc&oo:ch. Proceed"r.gs, 1984. Edi~ed by K. Neumann and D. Pallaschke. V. 190 pages. 1905. Vol. 241~ U. Wi~tma~n, Das Kenzep~ ratio ...~;;lIQr Preiscrwa.~ungen. XI, 310 Seiten. 1986. Vo:. 242: Decision Making wit'1 M;J~tiple Objec!ivcs. Proceedings l :984. Edited by Y. Y. Ha'm~s and V. C~ankorg. Xl. 57~ pages. 1985. Vo·. 243~ Integer Prcgrarr:n:rg ard Re~ated Arens. A C;assifioc Bibliography 1981-1984. Edited by R. von Rar.cow. XX, 386 paGes. 1985. Vol. 244~ Advances in Equilibrium ThE'ory. Proceed:ngs, 1984. Edited by C. O. Aliprantis. O. Burkinshnw and N. J. RO~h~an. U. 235 pages. 1985. Vol. 245~ J. E.M. Wil!"c:rr.. A~b;trage Theory. Vri. 114 pages. 1985.

Vel. 220; M. Aoki. Notes o~ Economic Time Series Analysis: S)'s~em Theoretic Perspectives. IX. 249 pages, 1983.

Vol. 246: P.W. Otter, :Jyna-nic FeatLire ScacQ ModeLing, Fl~ter;ng and S~lf-Tuning ControJ of Stocr.astic Systems. XiV, 177 pages.1985.

Vol. 221~ S. Nakamura, An rnter-lr.dJstry Translog Model of Prices ar.d Tecl"~ical C ha'1ge ~or the Wesi Gorman Economy. XIV. 290 pages. J984.

Vo!. 247: Optimization and Disc~ote C~cice ir Urban Syste:r.s. Proceedings. 1983. Edited by B.G. Hutchinsen. P. Nijkamp ar.d f,I.. Baty. VI, 371 pages. 1985.

vor. 222: Po J\/.eier. Einergy Systems. Analys;s for Deve!op:r.g Co':.! '1~rics. VI. 344 pages. 19B4. Vol. 22.3: W. Trockel. Markot Derr.e:nd. VIII. 205 pages.. 1984.

Vol. 248: Plura: Rafon~"ty and In~eraC~'o'e Decision Processes. Pro~ ceed;ngs. 1964. Edited by M. Grauo:". M. Thompson and A.P. Wierlbici
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Vol. 250~ M. Roube!',s. Ph. Vincke, Pro!c:enco MoCoiling. VIII.

Voi. 226: Selected Top' cs in Operations Research a~d Milt"le:r.at:cal Ecor.omics. Proc!:ed; ngs. 1983. Edited by G. Ha"mer a~d D. Pallaschke_ IX, 47S pages. 1984. Vol. 227: Risk a~Q Capital. Proeeed:ngs. 1903. Edited by G. Ba!r.berg and K. Spremann. VIt, 306 pages. 1984. Vol. 228: NOOllinearModels of f:;Jctua~ing Growth. Proceedings. 1983. Edited by R.M. Goodwin. fII. KrOger
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cr. Co~t:ngont

Clai~s. VIII. 151

Vol. 255: Nondifferer:tiab:e Optim'zatier~ Mot\'ations and Applications. Prcccedi:-:gs. 1984. Editod by V. F. Demy3t1ov a":.d D. Par
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Vol. 25B~ H. Funke. E~ne allgemeine Tneorio der Polypol- und Oli90polpreisbildu~g.III. 237 pages. 1985.

Va:. 233: G. Wagcnha~s. The World Copper Market X" 190 pages.

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Vol. 301: J. Kacprzyk. M. Rc....bc:-s (Eds.'. Non-Co:'lvontion.aJ Pre~or. onco Relations in Deo:s:on Mak:ng. VU, 155 P.1gCS. 1988... _ ....

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·Processcs. X. 323 pagos. 198'7. Vol.. :285: Toward Intcract:vo and Intel;;gent Dec'sion Support Syste..rs. Yolume 1. ProcecdEngs. 1986. Editod by Y. Sawaragi, K. lnouo and H. Nakayama.. XU} 445 pages. 1987. .

.vo~. 286~' Tovva'tl Jntoraet:vD and Int~I::ge.,t Decision Support Systoms. Vo'ume 2. J'roceedi:-gs, 7986. Ectitect by Y. Sawaragi, K. Inoue a.~d H. Nako.~aTa. X!J. 450 pages. 1987•._ Vol.' 287: Dyr.amicaJ Systems. Proceedings. 1985. Edited by A. B. KUr2hanski and K. ·Sigmund. VJ. 215 pages, 198.1. Vol. 2ea~ G.D. Rudebusch. Tho Es:imaticn of Macroac.cnomic D~s. equ;:ibr;um Mode!s with Reg~me Classif:catlon Information•. vrr, 128 pages. 19B'7. " -Yol. 289: B.R. Mo~boom. P;a"':ning in Decontralized Firms. X. 168 '·~es. '987.' ·Vol. ·290: C.A. Carlson. A. Haurio. 'XI. 264 pages. 198'7.

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Vol. 314: R.. T:e.tl.} W. Aloors. R. ~~,.~n ~E03.}, Bounced ~~~m Behavior in &por.~entaJ Games and Marke~ .• VJ. 368 pages. 1 ssa. Vol. 315: J. Orlshimo, G.l.O. Hewings. P. Nij~amp (Eds.). ·I~for.r.a-· (on Techr.ology: Social and Spatial PCfspeclves.. ;Proceed\~9s,.' 1966. VI. 268 pages. 1988. __

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Va:. 319: D. T. Luc, Thee;y of Vec!or Optim~.zatlon. V~II. ~ '73 pages. 1989. ' -

T. Vasko (Ed.)

The Long-Wave Debate Selected papers from an ITASA (lntemationa1lnstitute for Applied Systems Analysis) International Meeting on Long-Tenn Fluctuations in Economic Growth: Their Causes and Consequences) IIeid in Weimar, German Democratic Republic, June 10-14, 1985 1987.128 figures. XVII, 431 pages. ISBN 3-540-18164-4 Contents: Concepts and Theories on the Intc!?reration ofLong-Tcnn Fluctuations-in Ecor.omic Gro\\1h. - Technical Revolutions and Long Waves. - The Role of Financial and Monetary Variables in the LongWave Context. - I\-fodeling the Long-Wave Context. - :vIodc1ing the Long~Wave Phenomenon. - List of Participants. I. Boyd~ J. ~t Blatt

Investment Confidence and Business Cycles 1988. 160 pages. ISBN 3~540-18516~X Contents: Introduction and bricfsummary. - A brief historical survey of the trade cycle. - Literature on confidence. - The dominant theories. A first look at the new model. - Confidence. - Description of the model. The longer run. - Some general remarks. - Appendices. - References. Index. M. Faber. H. Niemes t G. Stephan

Entropy, Environment and Resources An Essay in Physico-Economics With the cooperation arL. Freytag Translated from the German by I. Pellengahr

1987. 33 figures. Approx. 210 pages.lSB~ 3~540 ..18248-9 The special features of the book are thal the authors utilize a natural scientific variable, entropy, to relate the economic system and the environment, that environmental protection and resource use arc analyzed in combination, and that a replacement of techniques over time is analyzed. A nove] aspect is that resource extraction is inteTJlrcted as a reversed diffusion process. Thus a relationship between entropy, energy and reM source concentration is estabHshed.

E.vanDamme

Stability and Perfection of Nash Equilibria 1987. 105 figures. Approx. 370 pages. ISBN 3 540-17101-0 Contents: Introduction. - Games in :KonnaJ Form. - ~Jatrix and Bimatrix Games. - Control Costs. - Incomplete Information. - Extensive POnTI Games. - Bargaining and Fair Division. - Repeated Games. - Evolu· tionary Game Theory. - Strategic Stability and Applications. - References. - Survey Diagrams. - Index. 8

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