Optimality conditions for maximizations of set-valued functions

JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS: Vol. 58, No. 1, JULY 1988

Optimality Conditions for Maximizations of Set-Valued Functions H. W.

CORLEY 1

Communicated by L. Cesari

The maximization with respect to a cone of a set-valued function into possibly infinite dimensions is defined, and necessary and sufficient optimality conditions are established. In particular, an analogue of the Fritz John necessary optimality conditions is proved using a notion of derivative defined in terms of tangent cones.

Abstract.

Key Words. Optimality conditions, set-valued functions, cones, tangent cones.

I. Introduction

In this paper, we define the maximization of a set-valued function with respect to a cone into possibly infinite dimensions and establish optimality conditions. Such problems have apparently not been explicitly studied heretofore, although in the duality theories of Tanino and Sawaragi (Ref. 1) and Corley (Ref. 2) for multiobjective programming the dual problems took this form. The main result here is an analogue of the Fritz John necessary optimality conditions of mathematical programming, which is stated in terms of derivatives for set-valued functions defined via tangent cones. Sufficient optimality conditions requiring a type of concavity are also given. Duality and existence are considered in Ref. 3. It is anticipated that an optimization theory for set-valued functions will provide a useful analytical tool because of the range of application of such functions. For example, Klein and Thompson (Ref. 4) survey their use in economics, in addition to presenting their theory. Zangwill (Ref. 5) uses them to present a unified treatment of convergence of nonlinear programming algorithms, while Hogan (Ref. 6) studies their properties from t Professor,Department of Industrial Engineering,Universityof Texas at Arlington, Arlington, Texas. 1 0022-3239/88/0700-0001506.00/0 ~ 1988 Plenum Publishing Corporation

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this viewpoint. Generalized equations (Ref. 7) and differential inclusions (Ref. 8) are other applications. Frequently occurring examples of set-valued functions include inverses of functions, cones of tangents, and subgradients.

2. Maximization of Set-Valued Functions Let X, Y, Z be real normed linear spaces, and let F : X ~ 2 r, G : X ~ 2 z be relations. In optimization literature, relations are often called set-valued functions, multifunctions, or point-to-set maps. We refer to them as setvalued functions. The domain of F : X ~ 2 Y is given by D o m ( F ) = {x c X : F(x)} ~ 0}. AsetKin

Yisaconeif

Ay~K,

for all y ~ K and A ~>0.

A pointed cone K is one for which K n - K = {0}. A convex cone K is one for which

hlyl+hEY2EK,

f o r a l l y l , Y E E K and A1, h2~>0.

The following notions o f optimality are used here. Let K be a pointed cone in Y and B C Y. For Yl, Y2 E Y, write Yl y',

[(x,.y,,)-(x',y')]/A,,->(x"-x',y"-y'),

SO it follows that ( x " - x ' , y " - y ' ) is in S [ G r ( F - K), (x', y')]. Thus, y " - y ' E CKF(x', y')(x"-x') to establish (4). []

JOTA: VOL. 58, NO. 1, JULY 1988

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4. Optimality Conditions for Problem (1) Differential optimality conditions for problem (1) are now developed. The necessary conditions in this and the next section are actually derived for weak maximal points, but they hold for maximal points since a maximal point is also a weak maximal point. Tangent cones apparently cannot distinguish between the two concepts, so it seems unlikely that stronger differential necessary can be developed for maximal points. The notation FA is used to denote the restriction of F to A.

Theorem 4.1.

I f Xo is a (weak) maximal point at Y0 for (1), then

CFA(Xo, yo)(X) n K ° = Q,

for all x e A,

and hence

DFA(Xo, yo)(X) n K ° = 0 ,

for all x ~ A.

Proof. To establish the first conclusion, suppose to the contrary that, for some ~ ~ A, there exists 33~ CFA(Xo, Yo)(X) n K °. Thus, 33# 0. By definition, (R, 33) c S[Gr(FA), (Xo, Y0)], and hence there are sequences (x,)CA, ( y , ) C Y, (t~n)C R 1, such that

xn~xo,

Y,~Yo,

y,~F(xn),

a,~O,

a, (xn - Xo, y, - Yo) -~ (x, 33). It follows that there exists N for which a , > 0 and an(y, -Yo) ~ K °C K\{O}, for n>~N. Since K is a cone, then y , - y o ~ K ° C K \ { O } , for n~>N. Thus, YN c F(xN) and YN - y o C K ° C K\{O}, in contradiction to Xo being a (weak) maximal point, and the first conclusion is established. The second follows immediately from the fact that DFA(Xo, Yo)(X) C CFA(Xo, Yo)(X) as a consequence of Result 3.1(c) and the definitions of the two derivatives. [] Sufficient conditions based on concavity are now stated for (1).

Theorem 4.2.

Let F be K-concave on the convex set A C D o m ( F ) . I f

K n DKF(xo, yo)(X-Xo) ={0},

for all x ~ A,

then Xo is a maximal point at Yo for (1). If

K° nDKF(Xo,Yo)(X-Xo)=Q,

for all x E A ,

then Xo is a weak maximal point at Yo for (1).

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Proof. Let x ~ A. It is easily established that G r ( F - K ) since F is K-concave, and hence

is convex,

DKF(xo, yo)(X) = CKF(xo, yo)(x), from Result 3.1(b) and the definition of the two derivatives. It now follows by hypothesis and Theorem 3.1 that, for all x e A, K c~ [ F ( x ) -Yo] C K n DrF(xo, yo)(X - Xo) = { 0}.

(5)

Thus, xo is a maximal point at Yo to (1); otherwise, for some x~ c A, there exists Yl ~ F(Xl) for which y~-yo~ K\{O}, in contradiction to (5). Using the cone K ° u {0} instead o f K in (5) similarly establishes the other conclusion. [3 Maximum-principle type necessary optimality criteria involving multiplier functionals can be readily obtained for (1) by applying to Theorem 4.1 a standard separation result; Theorem 4.2 also has an analogue. These results are stated below without proof. The following terminology is used. Let Y* denote the dual space of Y, and let K + = { / ~ Y*: l(y)>~O, for all y ~ K } denote the nonnegative dual cone of K. Then, l c K ÷ is definitely positive if l(y) > 0, for all y e K °, and strictly positive if l(y) > 0, for all y ~ K\{O}. We write 1F(x)