Efficiency and Proper Efficiency in Vector ... - Science Direct

JOURNAL

OF MATHEMATICAL

ANALYSIS

AND

93, 273-289 (1983)

APPLICATIONS

Efficiency and Proper Efficiency in Vector Maximization with Respect to Cones HAROLD P. BEN$ON* General

MoIors

Research Submitted

Laboratories,

Warren,

by Augustine

0. Esogbue

Michigan

48090

Received November 16, 1982

Vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region. The concepts of efficiency and proper efficiency have played a useful role in the analysis of such problems. Recently these concepts have been extended to vector maximization problems in which the underlying domination cone is a convex cone. In this paper, efftcient and properly efficient solutions for the vector maximization problem in which the underlying domination cone is any nontrivial, closed convex cone are examined. Differences between properly and improperly efftcient solutions are established. Characterizations of efficient and properly efficient solutions are presented, and conditions under which efficient solutions exist and fail to exist are derived.

1. INTRODUCTION The vector maximization problem involves the simultaneousmaximization of p > 2 noncomparable criterion functions over a given feasibility region. The concept of eflciency has played a useful role in analyzing this problem. In order to exclude certain efficient solutions which display an undesirable anomaly, a slightly restricted definition of efficiency, that of proper eflciency, was first proposed by Kuhn and Tucker [lo] and later refined by Geoffrion [8]. The underlying domination cone utilized in these definitions of efficiency and proper efficiency was the ordinary nonpositive orthant. Subsequently, the vector maximization problem in which the underlying domination cone is any convex cone was proposed for study. Yu [19] explored the geometry of the set of all efficient solutions for this problem under the assumption that the underlying domination cone is convex. He also proposed two procedures for actually generating the set of efficient solutions. Later, Borwein [4] proposed a definition of proper efficiency for the case * Present address: Department of Management and Administrative Sciences, University of Florida, Gainesville, Florida 32611.

273 0022-247x/83

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Copyright 0 1983 by Academx Press, Inc. All rights of reproduction in any form reserved.

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when the domination cone is any nontrivial, closed convex cone. Recently. Benson [2] has refined Borwein’s definition. In this paper, properties of efficient and properly efficient solutions for the vector maximization problem in which the underlying domination cone is any nontrivial, closed convex cone are developed. Many of the results shown here generalize earlier work which pertains to the case when the domination cone is the nonpositive orthant [ 1, 3, 7, 171. In Section 2 basic definitions and preliminary results are presented. In Section 3 the difference between properly and improperly efficient solutions, in terms of projecting cone relationships, is established. In addition, necessary and sufficient conditions for a point to be efficient and for an efficient point to be properly efficient are derived. In Section 4 conditions under which efficient solutions exist and fail to exist are developed. Conclusions are given in Section 5.

2. BASIC DEFINITIONS

AND PRELIMINARIES

Let S 5 RP be any nontrivial cone, where p > 2. Consider a vector-valued criterion function

defined over a set XC R”, where 4: X + R, Vj E J = { 1, 2,..., p}. The vector maximization problem VMAX:f(x)

subject to

x EX

P>

is the problem of finding all solutions that are efficient in the senseof DEFINITION

X E X and f(f)

A point X is said to be an eflcient solution of (P) if Ef(x) + S for some x E X implies that f(x) =f(Z).

2.1.

Let X, denote the set of all efficient solutions of problem (P). Following Yu [ 191, S will be referred to as the domination cone for (P). It is assumed throughout this paper that S is a nontrivial, closed convex cone, that X is a nonempty set, and that the interior of the dual cone for S, denoted (int S*), where

S*=

{s* ERP/(s*,s) 0 ] and, for any set C, let cl C denote the closure of C. Then the projecting cone of a set C is defined as follows: DEFINITION 2.2. Let C s RP. The projecting cone of C, denoted P(C), is the set of all points h of the form h = ty, where t E R + and 7 E C.

The projecting cone of a set C is also known as the cone generated by C and the conical extension of C. The projecting cone concept has been widely used in mathematical programming and convex analysis (for instance, see Canon et aE. ]5 J, Kelley and Namioka 191, and Wijsman [ 181). Let f(x) = LOX) I x E 4 and f(X) + S = {f(x) + s 1x E X and s E S}. Benson’s definition of a properly efficient solution of (P) [2] is DEFINITION 2.3. A point X is said to be a properly eflcient (P) when X E X, and cl P[f(X) + S - {f(x)}] n -S = {O}.

solution of

As pointed out by Benson (21 the motivation for considering only properly efficient solutions is to exclude any efficient solution X for which sequences {y’} in f(X) + S and {ti} in R, exist such that the directions hi = ti[ y’ -f(f)] h ave a nonzero limit which belongs to -S. Let XPRE denote the set of all properly efftcient solutions for (P). An efficient solution which is not properly efficient is said to be improperly efficient. For the case when S = RP , Definition 2.3 coincides with Geoffrion’s [8] definition of proper efftciency, as shown by Benson [2, Theorem 3.21. Two single-objective mathematical programming problems are central to the development of many of the results of this paper. Let kE X and so E (int S*). The first problem, h3.m

= suP(s”JIx))

subject to f(x) -f(f)

E -s,

x E x,

Pso,3

is a generalization of a class of problems frequently studied in the case when S = RP (see, for example, Benson [ 11, Charnes and Cooper (61, Ecker and Kouada [7], and Wendell and Lee [ 171). The second single-objective problem, max(s*,f(x))

subject to

x E X,

(P,*>

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where s* E S”, was studied by both Benson [ 21 and Borwein (4) for s* E (int S*). It, too, is a generalization of a problem class often studied for the case S = R!. (see Geoffrion 181 and Zeleny 1201, for example). To develop our prcperties of efficient and properly efficient solutions for (P), two types of preliminary results must be given. First, conditions under which problem (P,“,,_) is stably set need to be presented. Second, some basic properties of convex cones must be given. The remainder of this section is devoted to the presentation of these preliminary results. As a prerequisite to the presentation of conditions under which problem (P,o.i.) is stably set, we need to define some additional terms. Definition 2.4 is adapted from Luenberger [ 111. DEFINITION 2.4. Let g: Z-+ RP be a vector-valued function, where Z c R” is a nonempty convex set. Suppose Tc Rp is a cone. Then g is said to be a concave function with respect to T on Z when, for any a such that 0 ,< a < 1 and for any z ‘, z* E Z,

{g[az’+(l-a)z*I--ag(z’)-(l-a)g(z*)}ET. This definition is closely related to the more recent concept of cone convexity developed by Yu [ 191. If T E RP is a convex cone, then a set YE RP is said to be T-convex when Y + T is a convex set. It can easily be seen that when T is a convex cone, if g: Z + RP is a concave function with respect to T on the nonempty convex set Z, then g(Z) is -T-convex. The converse, however, need not hold. For any v E RP, let F(v) = (x E X IS(x) -f(f) E -S + v), where -S + v denotes (U E RP 1u = --s + v for some s E S). 2.5. Let so E (int S*) and X E X. The perturbation ~,~,,(v) associated with (Ps,,,) is given by DEFINITION

function

and is defined on the nonempty set V = (v E R” / F(v) # 0}. Notice for any so E (int S*) and X E X that 0 lies in the domain V of p,,,,-(v) and, hence rp,,,,(O) is either finite or + co. From Luenberger 111, p.2161, for any s’E(intS*) and XEX, iffis a concave function with respect to -S on the convex set X, then e?)so,:is a concave function on V. DEFINITION

2.6.

Let so E (int S*)

and iE

X. Suppose

that f

is a

VECTOR MAXIMIZATION

WITH CONES

277

concave function with respect to -S on the convex set X, and that rp,,,,(O) is finite. Then (Ps,,,) is said to be stably set when

is finite for all u E V. Notice that the limit in Definition 2.6 exists for all v E V since, from the assumptions, (~,~,z is a concave function on V and rp,,,,(O) is finite [ 15, Theorem 23.11. In words, under the assumptions of the definition, a consistent problem (Ps0,3 with finite supremum rp,,,(O) is stably set when the directional derivative of q, 0,*- at 0 in the direction v is finite for all u E I’. This concept of stability has been frequently studied in mathematical programming (see Peterson [ 121 and Rorkafellar [ 13, 141, for example). The term stably set was first introduced by Rockafellar [ 13, 141. The stability of (PsO,,_) will be related to the well-known concept of a Kuhn-Tucker vector, which is defined as DEFINITION 2.7. Let so E (int S*) and 2 E X. Assume q$!,- is finite. Then a Kuhn-Tucker vector for (P& is any vector s* E S* such that

Kuhn-Tucker

vectors are also referred to as Lagrange multipliers [ 11, 151.

A necessary and sufficient condition for (P,& to be stably set is given in Theorem 2.1. This theorem will be used both to prove a result in Section 3 which characterizes elements of XpRE and, in Section 4, to derive sufficient conditions for XPRE to be empty. The theorem is a direct consequence of a more general result of Rockafellar [ 15, Corollary 29.1.21. THEOREM 2.1. Assumef is a concave function with respect to 4 on the convex set X. Let so E (int S*) and X E X. Suppose cp+JO) is fmite. Then (Ps,,,3 is stably set if and onZyifa Kuhn-Tucker vector existsfor (P,,,).

As a final prerequisite to the development of our main results, three basic properties of convex cones will now be given. Throughout the remainder of this section, all cones are assumed to lie in RP. For any set T E RP, let T* denote the dual cone of T, let (ri 7’) denote the relative interior of T, and let T/(0} denote (t 1t E T, t # O}. The following properties of cones and dual cones are stated here for reference: (Pl) (P2)

For any set T E RP, T” is a closed convex cone. For any set TG RP, T* = (cl T)*.

HAROLD P.BENSON

278 (P3)

If T is a closed convex cone, then (i)

(int T*) = (t* / (t*, t) < 0, Vt E T/(0}},

(ii)

(int T) = {t 1(t, t*) < 0, Vt* E T*/(O)}.

and

PROPOSITION 2.1. Let C be a closed convex cone with (int C) # 0. Assume c E C and c’ E (int C). Then (c + c’) E (int C).

Proof

Immediate from (P3)(ii) and the definition of C*.

PROPOSITION 2.2. Let C be a convex cone and let T be a nontrivial closed convex cone with (int T*) # 0. Assume C f7 T = (0). Then [-(C*)n T*] # (0).

ProoJ Either 0 E (ri T) or 0 @(ri T). Suppose0 E (ri T). Let t E T/{O}. From ] 15, Theorem 6.41, there exists a B > 1 such that (1 - 8) t E T. Since T is a cone, -t = [l/(8-- l)](l - 0)t E T. Therefore, (Tn -T) # (0). However, since (int T*) # 0, it follows from (P3)(i) that (Tn -T) = {O}. From this contradiction, we have 0 @ (ri r). Together with the assumption that C n T = {O), this implies that (ri C) n (ri T> = 0. Since (ri C) and (ri T) are convex sets, by a basic separation theorem for convex sets ] 15, Theorem 11.31, there exists a u E RP/{O) such that sup@,u)

< fyc,

u).

1E7

Since 0 E T, 0 < SUP(I, u>,

(ET

and since 0 fGC, inf(c, u) < 0.

CCC

Therefore, 0 = supteT(t, u) = inf&c, (t,u) 0,

vc E c.

n T*]. Since u # 0, this completes the proof.

In a related result, Borwein 141 has shown that if C and T are closed convex cones such that (int T*) # 0 and Cn T= {0}, then [-(C*)n (int T*)] # 0. PROPOSITION 2.3. Let C be a convex cone. Assume co E C and c’ E (ri C). Then there exists a vector c E C and a positive scalar 1 such that

AC’ = (CO+ c).

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279

Proof: By [ 15, Theorem 6.41, since C is a nonempty convex set with co E C and c’ E (ri C), there exists a 8 > 1 such that (1 - 0) co + Bc’ E C. Therefore, for some c2 E C, Bc’ = ((3- 1) co + c2. Multiplying both sides of this equality by [l/(6l)], we obtain

[e/(e- i)] d =2 + p/(8- i)] 2. Since 8> 1, [e/(8-- l)] and [l/(f3- l)] are positive numbers. Therefore, 1i/(e - I)] 2 E c, SOthat c = [l/(0 - l)] c* and A = [e/(0 - l)] satisfy the properties called for.

3. CHARACTERIZATIONS

OF EFFICIENCY

In this section, necessary and sufficient conditions for efficiency and proper efficiency are derived by examining problem (Ps,,,). In addition, new necessary conditions for efficiency and improper efficiency are given. The first result helps to illuminate the difference, in terms of projecting cone relationships, between properly and improperly efficient solutions for P>* THEOREM

3.1. Zf X E X,, then cl P[f(X)

+ S - {f(Y)}]

n (ri -S) = 0.

Proof: From [ 15, pp. 44-451, we may assumethat (ri -S) = (int -S). Let 2 E X. Suppose that h E cl P[f(X) + S - If@)}] n (ri -S). By Definition 2.2, there exists a sequence {fn} of nonnegative real numbers, a sequence (x”} in X, and a sequence {s”) in S such that t,[S(x”) + S” f(z)] --f h. From the proof of Proposition 2.2, since (int(-S*)) # 0, 0 & (ri -S). Therefore, h # 0, so that there exists a positive integer fi such that I,, > 0 Vn > K Since (ri -S) is a cone, and (ri -S) = (int -S), f(x”) + s” -f(Z) E (ri -S) Vn > ri. Pick some n’ > fi. Then f(x”‘) + s”’ -f(2) E (ri -S), so that f(x”‘) -f(Z) = -(s”’ + s) for some s E (ri S). From [ 15, Theorem 6.11 and the fact that (ri S) is a cone, this implies that J-(x”‘) f(g) E (ri -S). Therefore,f(x”‘) -f(Z) E -S and, since 0 6?(ri -S),J(x”‘) # f(Z). Hence, x”’ E X, f(x) Ef(x”‘) + S, and f(x”‘) #f(Z). By Definition 2.1, X @X,. By the contrapositive, the proof is complete. If X is an improperly efficient solution for (P), then, by Definition 2.3, cl P[f(X) + S - {f(Z)}] n-S # {O}. But by Theorem 3.1, cl P[f(X) + S (f(x))]n(ri-S)=0. Th erefore, the only points -S has in common with c’ W(X) + s - u-(3 t 1 are relative boundary points of -S. The key difference between properly and improperly efficient points, then, is that cl m”(X) + s - Lml 1contains no nontrivial points of -S when X E XPRE, but when I is improperly efficient, although cl P[f(X) + S - (f(~)}]

280

HAROLD P.BENSON

contains no relative interior points of 4, it does intersect the relative boundary of -S at a point other than zero. This implies that an improperly efficient solution relative to the domination cone S is properly efficient relative to any nontrivial, closed convex domination cone S’ c (ri S) U {O). The next result gives a necessary and sufficient condition for a point to be efficient for (P). This condition is related to the single-objective maximization problem (P,,),,). Let Xz),, denote the set of all optima1 solutions for (P,,,.,_). THEOREM

(ii) (iii)

3.2.

(i)

Zf X E X,, then X E Xs.x, Vs" E (int S*).

some so E (int S”), then X E X,. 2 E X, ifand only $2 E X$,,for some so E (int S*). Zf X E X$,,for

In this case, X E X$,x, Vs" E (int S*). Proof

(i) Let soE (int S*). Since X E X,, X E X. Since 4 is a cone, = 0 E -S. Thus, X is feasible in (P,,,,_). SupposeX 65X$,x. Then there exists an x’ E X with f (x’) -f (2) E 4 and (so,f (x’)) > (so, f(2)). This implies that f(X) E f (x’) + S and f (x’) ff (.V), which contradicts that X E X,, completing the proof for (i).

f(X) -f(X)

(ii)

Let X E X. Assume X 6ZX,. Then there exists an x’ E X such that + S and f(x’) #f(T). H ence f(T) -f(x’) E S/{O). By (P3)(i), since S is a closed convex cone, (int S*) = {s” / (s”, s) < 0, Vs E S/(O)). By assumption, (int S*) # 0. Let so E (int S*). Then (s”,f (2) -f(x’)) < 0, so that (s’,f(x’)) > (so, f(2)). Since x’ E X and f (x’) -f (2) E -S, 2 t?i X$,,. By the contrapositive, (ii) is established.

f(Y) Ef(x’)

(iii)

This is immediate from (i) and (ii).

Notice that Theorem 3.2 does not require any concavity assumptions. It provides a test for the efficiency of X as long as (P$,,,) can be solved for some so E (int S*). When S = RP , previous results [ 1, 171 state that X E X, if and only if X E X2,-, where e E RP is the vector of ones. Since e E (int RP,) and, when S = RP , S* = RP+, this result is a special case of Theorem 3.2. Suppose 5! E X,. By Theorem 3.2(i), ZE X,*,,, for all so E (int S*). The next theorem shows that, under a concavity assumption on f, the question of whether or not X E X,,, can be answered by examining problem (P,,,,,). In this theorem, a necessary and sufficient condition for an efficient solution to be properly efficient is given. THEOREM 3.3. Assume f is a concave function with respect to -S on the convex set X. Suppose X E X,.

(i)

If(P,,,,)

is stably set for some so E (int S*), then X E X,,,.

VECTOR

(ii) (iii)

MAXIMIZATION

WITH

281

CONES

then (P$,,,) is stably set Vs” E (int S*). ,IfTEX,,,, I E XPRE ifand ody if (P,,,) is stably setfor some so E (int S*).

In this case, (P,O,a) is srably set Vs” E (int S*). ProoJ: Since 2 E X,, ,? E X$,,, Vs” E (int S*) by Theorem 3.2(i). Hence, v,~,~O) = (sO,f($), Vs” E (int S*).

(i) Suppose that there exists an so E (int S*) such that (P& is stably set. By Theorem 2.1 and Definition 2.7, there exists an s* E S* such that fpso,$(0)= sup [(s”,.f(x>~ + (s*J-(x) -.#-(~))I~ XEX

Since qsO,dO) = (s”,f(Z)),

this implies that

(s”,f(X>) > (sO,f(“q + (s*Lm> -f(-f>h

VXEX,

which, upon rearranging, yields that

(SO+ s*,f(q)

VXEX.

> (SO+ s*,f(x)),

This means that .? is an optimal solution for (P,,,+s.). By (Pl) and Proposition 2.1, (so + s*) E (int S*). From Benson [ 2, Theorem 4.11, ~~XPFcE~ (ii) Suppose that there exists an so E (int S*) such that (Ps,,) is not stably set. Since v,~,~O) = (s’,f(Z)), and (so&?)) < sup,,,[(sO,f(x)) + *, it follows by Theorem 2.1 and Definition 2.7 (s*,f(x) -f(-q)l. Vs” E s that

(s”.fW

< Epxl(s”J(x)) + (s*%/-(x)-./y-q>],

Vs” Es*.

Therefore, for each s* E S* there exists an x’ E X such that (s”Jw or equivalently,

< l(~“J-w>)

+ (s*Lfw>

-fW>)l,

such that (SO+ S*,f(g)

< (SO+ s*,f(x’)).

Hence, for each ,I > 0 and s* E S”, there exists an x’ E X such that @(s” + s*>,f(n>) < (qs” + s”>J(x’)). Let s’ E (int S*). By (Pl), S* is a convex cone. Applying Proposition 2.3, it follows that d’s’ = (so + C) for some ,I’ > 0 and SE S*. This means that

282

HAROLD P.BENSON

s’ = (I/A’)(s” + Q. Since l/A’ > 0 and FE S*, by the argument above, there exists an x” E X such that

((W’>(s” -t Gf(9) < ((vw~”

+ QfW’)),

or, equivalently, such that W,f(3)

< (s’,f(x”)).

Therefore, X is not an optimal solution for (P,,). Since s’ is an arbitrary element of (int S*), X6? X,,, 12, Theorem 4.21. By the contrapositive. (ii) is established. (iii)

This is immediate from (i) and (ii).

Corollary 3.3 specializes Theorem 3.3 to the case when S = RP . Consider the problem

subject to

where 1 E RP, and X E X. COROLLARY 3.3. Assumef is a concave function with respect to RP, on the convex set X. Suppose the domination conefor (P) is given by S = RP . Assume X E X,.

(i) (ii) (iii)

If (PA.,) is stably setfor some/1E (int RP,), then X E X,,, lf.fE XPRE, then (P,,,r) is stably set Vii E (int RP,). 2 E X,,,

.

if and only if (PA,,-) is stably setfor some1 E (int RP,).

In this case, (Pn,,) is stably set VA E (int RP,). Proof:

Apply the theorem with S = RP .

Necessary and sufficient conditions for an efficient point to be properly efticient have been developed previously for the case when S = R? [3]. Under the assumptionsof the corollary, these conditions imply that X E X,,, if and only if (PC,,,_)is stably set, where e E RP is the vector of cones. The corollary, then, can be viewed as a generalization of these conditions, since e is a particular element of (int RP,). Let soE (int S*), and supposef is a concave function with respect to -S on the convex set X. From Theorem 3.3, an efficient point 2 for (P) is

VECTOR

MAXIMIZATION

WITH

CONES

283

properly efficient if and only if (Ps,,,) is stably set. Stability may not be an easy property to demonstrate directly. However, whenever {x E X If(x) f(Z) E -S) satisfies any “constraint qualification” which insures that a Kuhn-Tucker vector exists at optimality, (P,,,) is stably set. For example, if S and X are polyhedral convex sets and, for eachj E J,fj is a linear function, then for any X E X, {x E X If(x) -f(x) E -4) satisfies a generalized KuhnTucker constraint qualification of Varaiya [ 16, Def. 3.31. Thus any efficient point of a linear vector maximization problem (P) with a polyhedral domination cone is properly efficient. As shown in [2], when f is concave with respect to -S and X is convex, if X is an improperly efficient solution for (P), there cannot exist an s* E (int S*) such that X is an optimal solution for (P,,). However, the next theorem states that X must be an optimal solution for (P,,) for some s* E as*/(o).

THEOREM 3.4. In addition to the assumptions in Theorem 3.3, assume that X is an improperly eflcient solution for (P). Then there exists an s* E &S*/(O) such that X is an optimal solution for (P,*).

ProoJ: From [ll, p. 2161, for any so E (int S*), psa,,- is a concave function on the nonempty set V. Therefore, V is a convex set. By definition, V =f(X) + S -f(f) = (v E RP If(,f) -f(x) + ZI E S for some x E X}. Suppose c E vf7 -S. Then there exists an x’ E X such that f(Z) -f(x’) + c E S, or equivalently, such thatf(f) -f(x’) = s - c for somes E S. Since s, -c E S, (s - c) E S. Therefore, x’ E X and f(Y) Ef(x’) + S. Since X E X, and (int S*) # 0, by Definition 2.1 and (P3)(i), s = c = 0. Hence, vn -S = PI* Consider the cone C = aV, where aV = {c 1c = CYU for somez, E V, (r > 0). By Rockafellar [ 151, since V is a convex set, C is a convex cone. Let c E C n -S. Supposec # 0. Then c = av for some a > 0 and v E V. Because c E -S and -S is a cone, c/a E -S. Since c/a E V, this contradicts that vn -S = (0). Therefore Cn -S = (0). Since X is an improperly efficient solution for (P), cl P[f(X) + S (f(x))]n-S#(O).LethEclP[f(X)+S-{(f(x)}]n-s,h#O.Then,by Definition 2.2, there exists a sequence (t,} of nonnegative real numbers, a sequence {x” } in X, and a sequence {s”} in S such that tn[f(x”) + sn f(.~?)] + h. This implies that h E cl C. By Proposition 2.2, there exists a nonzero vector in [-(C*) f7 -(S*)]. Choose any such vector, and let it be denoted -s*. Then s* E (C* n S*]/{O}. Since h E cl C, (s*, h)< 0 by (P2). Also, because h E -S, (s*, h) > 0. Therefore (s *, h) = (s*, -h) = 0. From (P3)(i), since -h E S/(O), s* E &S*. Furthermore, since (s*, c) < 0, Vc E C, (s*, u) < 0, 409/93/l

19

HAROLD P.BENSON

284

Q’z!E V. This implies that (s*,f(x) + s) < (s*,f(X)), Qx E X and s E S. Since 0 E S, (~*,f(x)) < (s*,f(,F)), Vx E X. Notice from the proof that the conclusion of Theorem 3.4 remains true if the assumption that f is a concave function with respect to -S on the convex set X is replaced by the weaker assumption that f(X) is S-convex. Theorem 3.4 can also be proven by using a result of Yu ] 19. Corollary 4.7(i)]. The proof offered here, however, demonstrates that any point s* which satisfies the conditions called for by the theorem must be normal to all nonzero elementsof cl P[S(X) + S - (f(f)}] n -S. Although the converse of Theorem 3.4 does not hold, by adapting a result of Yu ] 19, Lemma 4.4(ii)], one obtains that, even in the nonconcave case, if .f@ XPRE and ff is the unique optimal solution for (P,+) for some s* E as*, then X is an improperly efficient solution for (P), In the case when S = R” , if S is a concave function on the convex set X, Theorem 3.4 guarantees that given any improperly efficient solution 2 for (P), there exists a vector 1 E RP,/(O} with Jj = 0 for at least one j E .Z such that X is an optimal solution for the problem max \’ A,#)

subject to

x E X.

FJ

4. EXISTENCE

OF EFFICIENT

(PA1

SOLUTIONS

In this section conditions under which efficient solutions exist and fail to exist are presented. Each of the existence results involves an examination of problem (P$,,,). Special casesof these results when the domination cone is the ordinary nonpositive orthant can be found in [ 1, 7, 171. Throughout this section, assumethat 2 is an arbitrary fixed element of X. The first existence result gives a simple sufficient condition for X, to be nonempty. It states that any optimal solution x’ for (P,O,.r) for some so E (int S*) is an efficient solution for (P). THEOREM

Proof:

4.1.

Zfx’ E X,*,,,for someso E (int S*), then x’ E X,.

Either (a) f(x’)

=f(f)

or (b)f(x’)

#f(x).

Case 1. f(x’) =f(z). Then X E X,*,,, so that, by Theorem 3.2(ii), 2 E X, . Since f(x’) =f(,F), x’ E X, . Case 2. f(x’) #f(x). Suppose x’ & X,. Then, by Definition 2.1, there exists some x” E X such thatf(x’) Ef(x”) + S andf(x”) #f(x’). Therefore, f(x”) -f(x’) E -S/(0}. Since f(x’) -f(f) E -S and -S is a convex cone, [J(x”) -f(x’)] + [f(x’) -f(x)] E -S. Therefore, f(x”) --f(y) E -S. By

VECTOR

MAXIMIZATION

WITH

(P3)(i), since so E (int S*), (s’,f(x”) (~‘,f(x’)), which is a contradiction, Ps’r.d

285

CONES

-f(x’)) > 0. Therefore, (s’,f(x”)) > since x’ E X,*,,, and x” is feasible in

By Theorem 4.1, if X$,, # 0 for some so E (int S *), then X, # 0. Notice that no concavity assumptions are made in the theorem. For any so E (int S*), there are two possible ways in which an optimal solution can fail to exist for (Pso,,-). Either ~,~,, 0, there exists an x.’ E X such that (&f(x”)) > ti and f(x”) -f(Z) E -S. Therefore, given any M > 0, there exists an x”’ E X such that

(s;f(x”>

-f(f))

> A4

(4.1)

and j-(x”‘) -f(f)

E -Is.

(4.2)

Suppose so E (int S*). By (PI) and Proposition 2.3, we may choose an s* E S* and a A, > 0 such that 2,s’ = S+ s*. Then, for any x E X, (~osOJ(x> -f(f)) Pick any M > 0. From such that f(x”) -f(f) E Setting A4 = ION in the f(n) E -S and (s”,f(xM) For any so E (int S*), by

= (w-(x) -f(f))

+ (s*J(x)

-f(f)).

(4.3)

(4.1)-(4.3), it follows that there exists an x“ E X -S, and (~Oso,f(~M) -f(f)) > M. Suppose N > 0. above, there exists an x”’ E X such that f(x”) --f(Z)) > N. This implies that P~~,~(O)= +a~. consider the dual problem (Dso,,-) for (P,&, given

By the well-known weak duality w,~,JO) > (o&O). When strict ~,~,,(0) - P,~,~O) is said to exist. that pr,dO) = +co for some SE solutions for (P) to the existence

theorem (see [ 11, p. 2251, for example), inequality holds, then a duality gap The following theorem, under the condition (int S*), relates the existence of efficient of certain infinite duality gaps.

286

HAROLD P.BENSON

THEOREM 4.2. Assume q,,(O) = $00 for some SE (int S*). Then x’ E X is an efficient solution for (P) on@ if there is an infinite duality gap between (P,“,:) and (Dsoqi),Vs” E (int S”).

ProoJ:

By Lemma 4.1, q1,,,,(0) = +co, Vs” E (int S*). Since, for all so E

(int S*), sup(SO&))

> V,o,,(O),

XEX

it follows that s~p~~~(~~,f(x)) so E (int S*) and s* E S*,

= + 03, Vs” E (int S*). Therefore, for all

sup(SO+ s*,f(x)) x Ex

= +co

since, by (Pl) and Proposition 2.1, so + s* E (int S”) for each such so and s*. Therefore, for all so E (int S*), s* E S*, and X’E X,

sup [(s”,f(x)) + (s*,f(x> -fW)l XEX = sup [(SO+ s*,f(x>)l IEX

- (s*>.m)

= +a. This implies that for all soE (int S*) and ZE X,

If .?E X,, then, by Theorem 3.2(i), ZE X$,i, Vs” E (int S*). This implies that ~,~,~(0) = (sO,f(.?)), tis’ E (int S*). Since w,~,,(O) = +co Vs” E (int S”), the theorem is established. By Theorem 4.2, if, for each x E X there exists an so E (int S*) such that (P,,,,) and (D,,,,) do not have an infinite duality gap, then the unboundednessof (P,,) for some FE (int S*) guaranteesthat X, = 0. Wendell and Lee [ 17, Theorem 31 derive a result similar to Theorem 4.2 for the case when S=RP. Our final existence theorem shows that, in the concave case, when (P,,,,,) is unbounded for some soE (int S*), then no properly efficient solutions exist for (P). THEOREM 4.3. Assume f is a concave function with respect to -S on the convex set X. If q,.AO) = +a~ for some BE (int S*), then X,,, = 0.

CONES

287

Assume x’ E XPRE. Let so E (int S*). By x’ E x,, x’ E X,*,,$ and, thus, (s’,f(x’)) = ~So,x(0). finite. By Theorem 3.3(ii), since x’ E XPRE, (PSu,,,) Theorem 2.1, this implies that we may choose a s* E S* for (PSo,,,). By Definition 2.7,

Theorem 3.2(i), since Therefore I,,,, is is stably set. From Kuhn-Tucker vector

VECTOR

MAXIMIZATION

WITH

Proof

%0,x@) = sup ](s”J(x)>

+ @*J(x)

XEX

-fW)l.

is finite. By (PI) and Proposition 2.1, Therefore, SU~,~~[(S~ + s*,f(x))] (so + s*) E (int S*). From Lemma 4.1, since o,,,(O) = $03, we have that (oSo+,.,,-(0) = +co. By definition of yl,,+,*,,(O),

sup (SO+ s*,f(x))

2 rpso+s*,Fw

XEX

This implies that SUP,,~(S’ + s*,f(x))

SU~,,~(S~ + s*,f(x)) = +co, which is finite, thus completing the proof.

contradicts

that

For the case when S = RP, Benson [ 1] and Wendell and Lee [ 171 have shown that, under the assumptions of Theorem 4.3, if rp,,,(O) = +CQ, then X PRE= 0, where e E RP is the vector of ones. This result is a special case of Theorem 4.3. Notice that, under the concavity assumption of Theorem 4.3, if (o,,,(O) = +co for some SE (int S*), then, since XpRE = 0, any x’E X, is improperly efficient. In this case, by Theorem 4.2, there must be an infinite duality gap between (PSo,J and (DSu,?) for all so E (int S*). The results in this section can be used to investigate the question of whether or not efficient or properly efficient solutions exist for (P). Given any feasible solution X for (P), whenever (PS,,,) has an optimal solution x’ for some so E (int S*), then x’ E X,. If (Pr,,) is unbounded for some SE (int S*), then X, = 0 if, for each x E X, there exists an so E (int S*) such that (P,,,,) and (D,,,,) do not have an infinite duality gap. If f is a concave function with respect to -S on the convex set X, then the condition that os,x(0) = fco for some FE (int S*) implies that X,,, = 0, regardless of the existence or nonexistence of duality gaps. Therefore if, for some so E (int S*) and X E X, an optimal solution exists for (PS,,,) whenever cpSo,,(0) is finite, and if f is concave with respect to -S on the convex set X, then solving (PS,l.Y) either generates an element of X, or demonstrates that X PRE -- 0.

5.

CONCLUSIONS

The vector maximization problem in which the underlying domination cone is a convex cone has been proposed. The definition of proper efficiency

288

HAROLD

P. BENSON

has been extended to this problem via the projecting cone concept. This paper has examined efficient and properly efficient solutions for the vector maximization problem in which the underlying domination cone is any nontrivial, closed convex cone. Basic characterization and existence theorems for efftcient and properly efficient solutions have been presented. Many of these results generalize earlier work pertaining to the case when the domination cone is the ordinary nonpositive orthant. It seems possible to develop various other properties of efftciency and proper efftciency for the case when the domination cone is nontrivial, closed, and convex.

ACKNOWLEDGMENTS The author would like to express his gratitude to Professor Augustine 0. Esogbue for his most helpful suggestions. Thanks also go to the anonymous reviewers for their constructive comments concerning an earlier draft of this paper.

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15. R. T. ROCKAFELLAR, “Convex Analysis,” Princeton Univ. Press, Princeton, N.J., 1970. 16. P. P. VARAIYA, Nonlinear programming in Banach space, SIAM J. Appl. Math. 15 (1967), 284-293. 17. R. E. WENDELL AND D. N. LEE, Efftciency in multiple objective optimization problems, Math. Programming 12 (1977), 406414. 18. R. A. WIJSMAN, Convergence of sequences of convex sets, cones and functions, Bull. Amer. Marh. Sot. 70 (1964), 186-188. 19. P. L. Yu. Cone convexity, cone extreme points, and nondominated solutions in decision problems with multiobjectives, J. Optim. Theory Appl. 14 (1974), 319-377. 20. M. ZELENY. “Linear Multiobjective Programming,” Springer-Verlag, New York, 1974.