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Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of

In recent years, a great attention has been paid to set-valued optimization problems; many authors (see, e.g., [

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 [

Taa [

This paper is divided into three sections. In the first Section we collect some of the concepts required for the paper. Section

Let

Let

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

A solution

Let

a minimum solution for (

a weak minimum solution for (

Let

Using the above relations, Kuroiwa [

Let

a lower minimal

a lower weak minimal

In this way, the problems (

In these cases,

The next proposition supplies a characterization of l-w minimum (see [

There exists

Let us recall the following definition.

Let

The following definition has been introduced by Shi [

Let

Let

The set-valued derivatives

For sufficient condition for l-w minimal solution of problems

Let

We say that

Let

According to derived necessary condition, we recall the following notion of the strict l-w minimum and the concept of the

Let

A subset

The following Lemma has been established in [

Let

Suppose the contrary; then there exist

Necessary conditions for the problem

Let

Suppose the contrary; then there exist

As an immediate consequence we have the following corollary.

Let

Then,

It is obvious that

Another consequences of Theorem

Let

Let

Then

If

In the following we are going to prove necessary optimality conditions for

In the sequel the couple

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

If

Suppose the contrary; then, for every neighbourhood

Let us formulate necessary conditions for the problem

Let

Suppose the contrary; then there exist

As an immediate consequence we have the following corollary.

Let

Then

It is well known from vector optimization that we can derive sufficient condition under

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

Let

Let us show that, for every

On the other hand, we have that

As an immediate consequence we have the following corollary.

Under the setting of Theorem

From Theorem

Let

The following two corollaries of the above result are immediate.

Let

Let

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

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