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

and Applied Analysis 3 (ii) (0, 0) ∈ Gr(DF(x, y)) ⊂ Gr(SF(x, y)). (iii) DF(x, y) = SF(x, y) whenever the graph of F is convex inX × Y. For sufficient condition for l-wminimal solution of problems (SP1) and (SP2), we need certain convexity assumptions which are taken from [3, 4]. Definition 7. Let Γ ⊂ Y and (x, y) ∈ Gr(F). F is called Γcontingently quasiconvex at (x, y), if, for every x ∈ M, the condition (F(x)−y)∩Γ ̸ = 0 ensures thatDF(x, y)(x−x)∩Γ ̸ = 0. Definition 8. We say that F is KY-pseudoconvex at (x, y) ∈ Gr(F) if and only if F (x) − y ⊂ DF (x, y) (x − x) + KY. (11) Remark 9. Let Γ ⊂ Y. The Γ-contingent quasiconvexity reduces to theKY-pseudoconvexity. 3. Necessary Optimality Conditions According to derived necessary condition, we recall the following notion of the strict l-wminimum and the concept of theKY-wminimal property given in [10]. Definition 10. Let x be an l-wminimum solution of (SP1). x is called strict l-wminimum of (SP1), if there exists a neighbourhood U of x such that F(x) ̸ < F(x) for all x ∈ U ∩ M. Definition 11 (domination property). A subset A ⊂ Y has the KY-wminimal property if for all y ∈ A there exists a ∈Wmin(A) such that a − y ∈ (−int(KY)) ∪ {0}. Thefollowing Lemmahas been established in [10]without proof; we give a simple proof for reader’s convenience. Lemma 12. Let x, x ∈ M and y ∈ F(x). If F(x) ̸ < F(x), Wmin(F(x)) = {y}, andF(x) has theKY-wminimal property, one has (F (x) − y) ∩ (−int (KY)) = 0. (12) Proof. Suppose the contrary; then there exist y ∈ F(x) and k ∈ int(KY) such that y = y + k ∈ F (x) + int (KY) . (13) Since Wmin(F(x)) = {y} and F(x) have the KY-wminimal property we get y1 − y ∈ int (KY) ∪ {0} ∀y1 ∈ F (x) , (14) that is, y1 ∈ y + int (KY) ∪ {0} ∀y1 ∈ F (x) . (15) From (13) we have y1 ∈ F (x) + int (KY) , ∀y1 ∈ F (x) , (16) and hence F (x) ⊂ F (x) + int (KY) , (i.e., F (x) <F (x)) . (17) This contradicts F(x) ̸ < F(x). Necessary conditions for the problem (SP1) are given in the following. Theorem 13. Let x be a strict l-wminimum of (SP1). If Wmin(F(x)) = {y} and F(x) has theKY-wminimal property, then SF (x, y) (x) ∩ (−int (KY)) = 0, ∀x ∈ M. (18) Proof. Suppose the contrary; then there exist x ∈ M and y ∈ Y such that y ∈ SF (x, y) (x) ∩ (−int (KY)) , (19) and hence there exist (tn)n∈N > 0 and (xn, yn) → (x, y) such that tnxn 󳨀→ 0, tnyn ∈ F (x + tnxn) − y, ∀n ∈ N, (20) and from the hypothesis we have that x is a strict lwminimum of (SP1); then there exists a neighbourhoodU of x such that F(x) ̸ < F(x) for all x ∈ U∩M. Since x+tnxn → x then there exists n0 ∈ N such that x + tnxn ∈ U ∩ M for all n ≥ n0 and then F (x + tnxn) ̸ <F (x) , ∀n ≥ n0, (21) by Lemma 12 and hypothesis we get [F (x + tnxn) − y] ∩ (−int (KY)) = 0, ∀n ≥ n0. (22) On the other hand, y ∈ −int(KY); then there exists n1 ∈ N such that tnyn ∈ −int (KY) , ∀n ≥ n1, (23) and for (20) we have [F (x + tnxn) − y] ∩ (−int (KY)) ̸ = 0 ∀n ≥ n1. (24) This contradicts (22) for all n ≥ max(n0, n1). As an immediate consequence we have the following corollary. Corollary 14. Let x be an lwminimum of (SP1). Let y ∈ F(x). Let us suppose that there exists a neighbourhood U of x such that for each x ∈ U∩M one of the following conditions is satisfied: (a) y ∉ F(x) + int(KY) or (b) y ∈ Wmin(F(x), KY). 4 Abstract and Applied Analysis Then, (i) x is a strict l-wminimum of (SP1); (ii) SF(x, y)(x) ∩ (−int(KY)) = 0, for all x ∈ M. Proof. It is obvious that (b) ⇒ (a) ⇒ F(x) ̸ < F(x). Then if there exists a neighbourhood U of x such that for each x ∈ U∩M the condition (a) holds, we deduce from Definition 10 that (i) holds. On the other hand if (a) holds we have (F (x) − y) ∩ (−int (KY)) = 0, ∀x ∈ U ∩M. (25) By using similar arguments as inTheorem 13, we establish (ii). Another consequences of Theorem 13 and Corollary 14 are given in the following corollaries. Corollary 15. Let x be a strict l-wminimum of (SP1). If F(x) ̸ < F(x), Wmin(F(x)) = {y}, and F(x) has the KYwminimal property, then DF (x, y) (x) ∩ (−int (KY)) = 0, ∀x ∈ M. (26) Corollary 16. Letx be an l-wminimumof (SP1). Lety ∈ F(x). Let us suppose that there exists a neighbourhood U of x such that for each x ∈ U ∩ M one of the following conditions is satisfied: (a) y ∉ F(x) + int(KY) or (b) y ∈ Wmin(F(x), KY). Then (i) x is a strict l-wminimum of (SP1); (ii) DF(x, y)(x) ∩ (−int(KY)) = 0, for all x ∈ M. Remark 17. If x is a strict l-wminimum solution of (SP1) and y ∈ Wmin(F(x)), 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 Rodŕıguez-Maŕın [10]: let F : (0, 2) 󴁂󴀱 R be defined by F (x) = { { { {(y, z) ∈ R | (y − 2x + 2)2 + z ≤ x} , if x ≤ 1, {(y, z) ∈ R | (y − 2x + 2)2 + z ≤ (2 − x)} , if x > 1. (27) Let KY = R2+. Then x = 1 is a strict l-wminimum of F and y = (−1, 0) ∈Wmin(F(1)). But observe that Wmin(F(x)) ̸ = {y} and for all neighbourhoodsU of x there exists x ∈ U∩M such y ∈ F(x) + int(KY) and y ∉Wmin(F(x)). On the other hand for all k ∈ −int(KY) and x ∈ M we take tn = 0 for each n ∈ N, then for every sequence (xn, yn) → (x, k) we have (−1, 0) + tnyn ∈ F (1 + tnxn) , ∀n ∈ N, (28) that is DF (1, (−1, 0)) (x) ∩ (−int (KY)) ̸ = 0, ∀x ∈ M. (29) Hence SF (1, (−1, 0)) (x) ∩ (−int (KY)) ̸ = 0, ∀x ∈ M. (30) In the following we are going to prove necessary optimality conditions for (SP2) in terms of contingent derivative. In the sequel the couple (F, G) is a set-valued map from X into Y × Z defined by (F, G) (x) = (F (x) , G (x)) . (31) Let z ∈ G(x) ∩ (−KZ) and we consider the following problem (SP3) with respect to KY × (KZ + z): l-minimize (F, G) (x) , subject to x ∈ M. (SP3) The following result compares the set of strict lwminimum solution of (SP2) to the set of strict l-wminimum solution of (SP3). Proposition 18. If x is a strict l-wminimal solution of (SP2) then for all z ∈ G(x)∩(−KZ), x is a strict l-wminimal solution of (SP3) with respect to KY × (KZ + z), Proof. Suppose the contrary; then, for every neighbourhood U of x, there exists x ∈ U ∩M such that (F, G) (x) ⊂ (F, G) (x) + int (KY × KZ + z) , (32)


Introduction
In recent years, a great attention has been paid to set-valued optimization problems; many authors (see, e.g., [1][2][3][4][5][6][7]) have concentrated on the problems with and without constraints: minimize  () ,  ∈ , minimize  () , 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.
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 dom () fl { ∈  |  () ̸ = 0} , Let  :    be a set-valued map and let us suppose that dom  = dom  =  with  ̸ = 0. Research in set-valued optimization has concentrated on the problems with and without constraints: minimize  ()  ∈ . ( minimize  () 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 () = ⋃ ∈ ().
(ii) a weak minimum solution for (3) and we denote  ∈ W min(,   ) (or  ∈ W min()), 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 lminimal set, resp.).
≤   imply  ≤  .The family of l-minimal sets of S is denoted by l-min(S,   ) (or l-min(S)); (ii) a lower weak minimal (or l-w minimal) set of S, if  ∈ S and  <   imply  <  .The family of weakly l-minimal sets of S is denoted by l-W min(S,   ) (or l-W min(S)).
In this way, the problems ( 3) and ( 4) can be written in set optimization with the following forms: l-minimize  ()  ∈ .
(SP 1 ) In these cases,  is a l-minimum (lw minimum, resp.)solution of , if  ∈  (with () ∩ −  ̸ = 0 in the problem (SP 2 )) 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 lw minimum (see [10,Proposition 18]).
Let us recall the following definition.
For sufficient condition for l-w minimal solution of problems (SP 1 ) and (SP 2 ), we need certain convexity assumptions which are taken from [3,4].

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 (SP 1 ). is called strict l-w minimum of (SP 1 ), if there exists a neighbourhood  of  such that () ̸ <  () for all  ∈  ∩ .
The following Lemma has been established in [10] without proof; we give a simple proof for reader's convenience.
Necessary conditions for the problem (SP 1 ) are given in the following.
Theorem 13.Let  be a strict l-w minimum of (SP 1 ).If W min(()) = {} and () has the   -w minimal property, then Proof.Suppose the contrary; then there exist  ∈  and  ∈  such that and hence there exist (  ) ∈N > 0 and (  ,   ) → (, ) such that and from the hypothesis we have that  is a strict lw minimum of (SP 1 ); then there exists a neighbourhood  of  such that () ̸ <  () for all  ∈ ∩.Since +    →  then there exists  0 ∈ N such that  +     ∈  ∩  for all  ≥  0 and then by Lemma 12 and hypothesis we get On the other hand,  ∈ −int(  ); then there exists and for (20) we have This contradicts (22) for all  ≥ max( 0 ,  1 ).
As an immediate consequence we have the following corollary.
Proof.It is obvious that (b) ⇒ (a) ⇒ () ̸ <  ().Then if there exists a neighbourhood  of  such that for each  ∈  ∩  the condition (a) holds, we deduce from Definition 10 that (i) holds.On the other hand if (a) 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.
Let us suppose that there exists a neighbourhood  of  such that for each  ∈  ∩  one of the following conditions is satisfied: (a)  ∉ () + int(  ) or (b)  ∈  min((),   ).
Remark 17.If  is a strict l-w minimum solution of (SP 1 ) and  ∈ W min(()), 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  : (0, 2)  R 2 be defined by Let   = R 2 + .Then  = 1 is a strict l-w minimum of  and  = (−1, 0) ∈ W min((1)).But observe that W min(()) ̸ = {} and for all neighbourhoods  of  there exists  ∈  ∩  such  ∈ () + int(  ) and  ∉ W min(()).On the other hand for all  ∈ −int(  ) and  ∈  we take   = 0 for each  ∈ N, then for every sequence (  ,   ) → (, ) we have that is In the following we are going to prove necessary optimality conditions for (SP 2 ) in terms of contingent derivative.
Let us formulate necessary conditions for the problem (SP 2 ).In the sequel we consider the following subset of :