Necessary and Sufficient Conditions for Set-Valued Maps with Set Optimization

Abstract

Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of -derivative and contingent derivative. Sufficient conditions for the existence of solutions are shown for set-valued maps under generalized quasiconvexity assumptions.

1. Introduction

In recent years, a great attention has been paid to set-valued optimization problems; many authors (see, e.g., [1–7]) have concentrated on the problems with and without constraints: where and are set-valued maps defined between two Banach spaces , and , , respectively, is the pointed closed convex cone of , and is a nonempty subset of .

Studies on these problems consider two types of solutions: vector solution, given by a vector optimization, and set solution, given by a set optimization.

The vector solution cannot be often used in practice, since it depends only on special element of image set of solution and the other elements are ignored; therefore the solution concept in vector optimization is sometimes improper. In order to avoid this drawback, Kuroiwa [8] introduced in the first time the concept of set solution by using practically relevant order relations for sets. This leads to solution concepts for set-valued optimization problems based on comparisons among values of the set-valued objective map. Hernández et al. [9] gives some links between solutions concepts in vector and set optimization.

Taa [7] gives necessary and sufficient conditions for unconstraint vector optimization in terms of -derivatives. Jahn and Khan [3] establish optimality conditions for unconstrained vector optimization under generalized convexity assumptions. Alonso-Durán and Rodríguez-Marín [10] give optimality conditions for the considered problems in set optimization using directional derivatives under pseudoconvexity assumptions and with the notion of the contingent derivative. In this paper we study necessary conditions for both problems in terms of -derivatives with set optimization and we derive sufficient conditions under weaker notion of pseudoconvexity assumptions that are given in [3].

This paper is divided into three sections. In the first Section we collect some of the concepts required for the paper. Section 2 is devoted to the necessary optimality conditions for the unconstrained and the constrained set optimization and Section 3 deals with the sufficient optimality conditions in set optimization.

2. Preliminaries

Let , , and be real normed spaces, where and are partially ordered by convex pointed cones with nonempty interiors and , respectively. and will denote the continuous duals of and , respectively. The collection of nonempty subsets of will be denoted by

Let be a set-valued map. We recall that the effective domain and the graph of are defined by Let be a set-valued map and let us suppose that with

Research in set-valued optimization has concentrated on the problems with and without constraints:

A solution for these problems with the criterion of vector optimization is defined as a generalization of the notion established by Pareto. We recall this concept in the following definition. Let .

Definition 1. Let . It is said that is (i)a minimum solution for (3) and we denote   or , if there exists such that (ii)a weak minimum solution for (3) and we denote (or ), if there exists such that

Let (, resp.) be the following relation defined between two nonempty subsets , of :

Using the above relations, Kuroiwa [8], in a natural way, introduced the following notion of l-minimal set weakly l-minimal set, resp.).

Definition 2. Let . It is said that is (i)a lower minimal or l-minimal set of , if and imply . The family of l-minimal sets of is denoted by - (or l-);(ii)a lower weak minimal or l- w minimal set of , if and imply . The family of weakly l-minimal sets of is denoted by l-W  (or l-W ).

In this way, the problems (3) and (4) can be written in set optimization with the following forms:

In these cases, is a l-minimum l- w minimum, resp. solution of , if   with in the problem ) and is a l-minimal l-w minimal, resp. set of the family of images of , that is, the family

The next proposition supplies a characterization of l-w minimum (see [10, Proposition  18]).

Proposition 3. is an l-w minimal solution of if and only if for each one of the following conditions is satisfied: (i) and (ii)There exists such that

Let us recall the following definition.

Definition 4. Let . The contingent derivative is the set-valued map from into defined by if there exist sequences , such that

The following definition has been introduced by Shi [6]. It is an extension of the set-valued derivative in Definition 4.

Definition 5. Let . The -derivative is the set-valued map from into defined by if there exist sequences , such that

Remark 6. Let . It is easy to see the following: (i)The set-valued derivatives and are positively homogeneous with closed graphs.(ii)(iii) whenever the graph of is convex in

For sufficient condition for l-w minimal solution of problems and , we need certain convexity assumptions which are taken from [3, 4].

Definition 7. Let and . is called -contingently quasiconvex at , if, for every , the condition ensures that

Definition 8. We say that is -pseudoconvex at if and only if

Remark 9. Let . The -contingent quasiconvexity reduces to the -pseudoconvexity.

3. Necessary Optimality Conditions

According to derived necessary condition, we recall the following notion of the strict l-w minimum and the concept of the -w minimal property given in [10].

Definition 10. Let be an l-w minimum solution of . is called strict l-w minimum of , if there exists a neighbourhood of such that for all

Definition 11 (domination property). A subset has the -w minimal property if for all there exists such that

The following Lemma has been established in [10] without proof; we give a simple proof for reader’s convenience.

Lemma 12. Let and . If , , and has the -wminimal property, one has

Proof. Suppose the contrary; then there exist and such thatSince W  and have the -w minimal property we get that is, From (13) we have and hence This contradicts

Necessary conditions for the problem are given in the following.

Theorem 13. Let be a strict l-w minimum of . If W  and has the -w minimal property, then

Proof. Suppose the contrary; then there exist and such that and hence there exist and such thatand from the hypothesis we have that is a strict l-w minimum of ; then there exists a neighbourhood of such that for all . Since then there exists such that for all and then by Lemma 12 and hypothesis we getOn the other hand, ; then there exists such that and for (20) we have This contradicts (22) for all

As an immediate consequence we have the following corollary.

Corollary 14. Let be an l- w minimum of . Let . Let us suppose that there exists a neighbourhood of such that for each one of the following conditions is satisfied: (a) or(b) Then, (i) is a strict -w minimum of ;(ii)

Proof. It is obvious that . Then if there exists a neighbourhood of such that for each the condition holds, we deduce from Definition 10 that holds. On the other hand if holds we have By using similar arguments as in Theorem 13, we establish (ii).

Another consequences of Theorem 13 and Corollary 14 are given in the following corollaries.

Corollary 15. Let be a strict l-w minimum of . If , W , and has the -w minimal property, then

Corollary 16. Let be an l-w minimum of . Let . Let us suppose that there exists a neighbourhood of such that for each one of the following conditions is satisfied: (a) or(b) Then (i) is a strict -w minimum of ;(ii)

Remark 17. If is a strict l-w minimum solution of and , Theorem 13 and Corollaries 14 and 16 are not guaranteed if the other conditions are not satisfied. Indeed, let us recall the example considered in Alonso-Durán and Rodríguez-Marín [10] let be defined by Let . Then is a strict l-w minimum of and . But observe that and for all neighbourhoods of there exists such and . On the other hand for all and we take for each , then for every sequence we have that is Hence

In the following we are going to prove necessary optimality conditions for in terms of contingent derivative.

In the sequel the couple is a set-valued map from into defined by Let and we consider the following problem with respect to :

The following result compares the set of strict l-w minimum solution of to the set of strict l-w minimum solution of .

Proposition 18. If is a strict l-w minimal solution of then for all , is a strict -w minimal solution of with respect to ,

Proof. Suppose the contrary; then, for every neighbourhood of , there exists such that then, and since , we get Thus for every neighbourhood of there exists such that This contradicts is a strict l-w minimal solution of .

Let us formulate necessary conditions for the problem . In the sequel we consider the following subset of :

Theorem 19. Let be a strict -w minimum solution of and . If has the -w minimal property and , then

Proof. Suppose the contrary; then there exist and such that and hence there exist and such thatOn the other hand, , then there exists such that and hence as there exists such that for every , then . Let , by (39) we getFrom hypothesis we have is a strict l-w minimum solution of ; then there exists a neighbourhood of such that for all . Since then there exists such that for all ; thus for all ; hence, and, by Lemma 12 and hypothesis, we get This contradicts (42).

As an immediate consequence we have the following corollary.

Corollary 20. Let be an -w minimum of and . Let . Let us suppose that there exists a neighbourhood of such that for each one of the following conditions is satisfied: (a) or(b) Then (i) is a strict -w minimum of ;(ii)

4. Sufficient Optimality Conditions

It is well known from vector optimization that we can derive sufficient condition under -contingently quasiconvex assumptions. Next, we establish sufficient condition with similar assumptions for set optimization. The following terminology is used. Let denote the dual space of , and let denote the nonnegative dual cone of

The next theorem provides a sufficient condition for the l-w minimum solution of .

Theorem 21. Let and . Assume that there are and such that andIf is -contingently quasiconvex at with then is an -w minimal solution of on

Proof. Let us show that, for every ,Assume the contrary; then there exist and such that Since , , and , we have This contradicts (46).
On the other hand, we have that is -contingently quasiconvex at ; thus (48) ensures that there is no such that that is Hence for every there exists such that by Proposition 3, and we deduce that is a l-w minimal solution of on

As an immediate consequence we have the following corollary.

Corollary 22. Under the setting of Theorem 21, if the map is -pseudoconvex at then is a -w minimal solution of on

From Theorem 21, we obtain the following sufficient optimality condition for .

Theorem 23. Let and . Assume that there exists such that If is -contingently quasiconvex at with, , then is an -w minimal solution of .

The following two corollaries of the above result are immediate.

Corollary 24. Let and . Assume that If is -contingently quasiconvex at with, , then is an l-w minimal solution of .

Corollary 25. Let and . Assume that If is -pseudoconvex at , then is an l-w minimal solution of .

5. Conclusions

This paper deals with a set-valued optimization problem which involves a set-valued objective and set-valued constraints. Since such problems involve set-valued maps, optimality conditions are often given using various notions of set-valued derivatives. In this paper, we use the notion of the so-called -derivative (and also the contingent derivative) to give necessary optimality conditions for the considered problems. For the sufficient optimality conditions, certain generalized notion of convexity is employed.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.