AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi 10.1155/2018/5962049 5962049 Research Article Necessary and Sufficient Conditions for Set-Valued Maps with Set Optimization http://orcid.org/0000-0003-4418-4406 Oussarhan Abdessamad 1 http://orcid.org/0000-0002-9042-0986 Daidai Ikram 1 Wong Patricia J. Y. Faculté des Sciences et Techniques Université Cadi Ayyad BP 549 Marrakech Morocco uca.ma 2018 112018 2018 15 08 2017 10 12 2017 112018 2018 Copyright © 2018 Abdessamad Oussarhan and Ikram Daidai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Optimality conditions are studied for set-valued maps with set optimization. Necessary conditions are given in terms of S-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., ) have concentrated on the problems with and without constraints: (1)minimizeFx,xM,minimizeFx,Gx-KZ,xM, where F and G are set-valued maps defined between two Banach spaces X, Y and X, Z, respectively, KZ is the pointed closed convex cone of Z, and M is a nonempty subset of X.

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  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.  gives some links between solutions concepts in vector and set optimization.

Taa  gives necessary and sufficient conditions for unconstraint vector optimization in terms of S-derivatives. Jahn and Khan  establish optimality conditions for unconstrained vector optimization under generalized convexity assumptions. Alonso-Durán and Rodríguez-Marín  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 S-derivatives with set optimization and we derive sufficient conditions under weaker notion of pseudoconvexity assumptions that are given in .

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 X, Y, and Z be real normed spaces, where Y and Z are partially ordered by convex pointed cones with nonempty interiors KY and KZ, respectively. Y and Z will denote the continuous duals of X and Y, respectively. The collection of nonempty subsets of Y will be denoted by (Y).

Let F:XY be a set-valued map. We recall that the effective domain and the graph of F are defined by (2)domFxXFx,GrFx,yX×YyFx.Let G:XZ be a set-valued map and let us suppose that domF=domG=M with M.

Research in set-valued optimization has concentrated on the problems with and without constraints:(3)minimizeFxxM.(4)minimizeFxGx-KZ,xM.

A solution x¯M 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 F(M)=xMF(x).

Definition 1.

Let x¯M. It is said that x¯ is

a minimum solution for (3) and we denote x¯min(F,KY)  (or x¯min(F)), if there exists y¯F(x¯) such that (5)FM-y¯-KY=0;

a weak minimum solution for (3) and we denote x¯Wmin(F,KY) (or x¯Wmin(F)), if there exists y¯F(x¯) such that (6)FM-y¯-intKY=.

Let l (<l, resp.) be the following relation defined between two nonempty subsets A, B of Y: (7)AlBBA+KY,resp.A<lBBA+intKY.

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

Definition 2.

Let S(Y). It is said that AS is

a lower minimal (or l-minimal) set of S, if BS and BlA imply AlB. The family of l-minimal sets of S is denoted by l-min(S,KY) (or l-min(S));

a lower weak minimal (or l- w minimal) set of S, if BS and B<lA imply A<lB. The family of weakly l-minimal sets of S is denoted by l-W min(S,KY) (or l-W min(S)).

In this way, the problems (3) and (4) can be written in set optimization with the following forms: SP1l-minimizeFxxM.SP2l-minimizeFxGx-KZ,xM.

In these cases, x¯ is a l-minimum (l- w minimum, resp.) solution of F, if x¯M  (with G(x¯)-KZ in the problem SP2) and F(x¯) is a l-minimal (l-w minimal, resp.) set of the family of images of F, that is, the family (8)F=FxxM.

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

Proposition 3.

x ¯ M is an l-w minimal solution of SP1 if and only if for each xM one of the following conditions is satisfied:

F(x)<lF(x¯) and F(x¯)<lF(x).

There exists y¯F(x¯) such that (F(x)-y¯)(-int(KY))=.

Let us recall the following definition.

Definition 4.

Let (x¯,y¯)Gr(F). The contingent derivative DF(x¯,y¯) is the set-valued map from X into Y defined by yDF(x¯,y¯)(x) if there exist sequences (tn)0+, (xn,yn)(x,y) such that (9)y¯+tnynFx¯+tnxn,nN.

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

Definition 5.

Let (x¯,y¯)Gr(F). The S-derivative SF(x¯,y¯) is the set-valued map from X into Y defined by ySF(x¯,y¯)(x) if there exist sequences (tn)]0,+[, (xn,yn)(x,y) such that (10)tnxn0,y¯+tnynFx¯+tnxn,nN.

Remark 6.

Let (x¯,y¯)Gr(F). It is easy to see the following:

The set-valued derivatives DF(x¯,y¯) and SF(x¯,y¯) are positively homogeneous with closed graphs.

(0,0)Gr(DF(x¯,y¯))Gr(SF(x¯,y¯)).

DF(x¯,y¯)=SF(x¯,y¯) whenever the graph of F is convex in X×Y.

For sufficient condition for l-w minimal solution of problems SP1 and SP2, we need certain convexity assumptions which are taken from [3, 4].

Definition 7.

Let ΓY and (x¯,y¯)Gr(F). F is called Γ-contingently quasiconvex at (x¯,y¯), if, for every xM, the condition (F(x)-y¯)Γ ensures that DF(x¯,y¯)(x-x¯)Γ.

Definition 8.

We say that F is KY-pseudoconvex at (x¯,y¯)Gr(F) if and only if (11)Fx-y¯DFx¯,y¯x-x¯+KY.

Remark 9.

Let ΓY. The Γ-contingent quasiconvexity reduces to the KY-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 KY-w minimal property given in .

Definition 10.

Let x¯ be an l-w minimum solution of SP1. x¯ is called strict l-w minimum of SP1, if there exists a neighbourhood U of x¯ such that FxlF(x¯) for all xUM.

Definition 11 (domination property).

A subset AY has the KY-w minimal property if for all yA there exists aWmin(A) such that a-y(-int(KY))0.

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

Lemma 12.

Let x,x¯M and y¯F(x¯). If F(x)lF(x¯), Wmin(F(x¯))={y¯}, and F(x¯) has the KY-wminimal property, one has (12)Fx-y¯-intKY=.

Proof.

Suppose the contrary; then there exist yF(x) and kint(KY) such that(13)y¯=y+kFx+intKY.Since W min(F(x¯))={y¯} and F(x¯) have the KY-w minimal property we get (14)y1-y¯intKY0y1Fx¯, that is, (15)y1y¯+intKY0y1Fx¯. From (13) we have (16)y1Fx+intKY,y1Fx¯,and hence (17)Fx¯Fx+intKY,i.e.,Fx<lFx¯. This contradicts F(x)lF(x¯).

Necessary conditions for the problem SP1 are given in the following.

Theorem 13.

Let x¯ be a strict l-w minimum of SP1. If W min(F(x¯))={y¯} and F(x¯) has the KY-w minimal property, then (18)SFx¯,y¯x-intKY=,xM.

Proof.

Suppose the contrary; then there exist xM and yY such that (19)ySFx¯,y¯x-intKY, and hence there exist tnnN>0 and (xn,yn)(x,y) such that(20)tnxn0,tnynFx¯+tnxn-y¯,nN,and from the hypothesis we have that x¯ is a strict l-w minimum of SP1; then there exists a neighbourhood U of x¯ such that F(x)lF(x¯) for all xUM. Since x¯+tnxnx¯ then there exists n0N such that x¯+tnxnUM for all nn0 and then (21)Fx¯+tnxnlFx¯,nn0, by Lemma 12 and hypothesis we get(22)Fx¯+tnxn-y¯-intKY=,nn0.On the other hand, y-int(KY); then there exists n1N such that (23)tnyn-intKY,nn1, and for (20) we have (24)Fx¯+tnxn-y¯-intKYnn1. This contradicts (22) for all nmax(n0,n1).

As an immediate consequence we have the following corollary.

Corollary 14.

Let x¯ be an l- w minimum of SP1. Let y¯F(x¯). Let us suppose that there exists a neighbourhood U of x¯ such that for each xUM one of the following conditions is satisfied:

y¯F(x)+int(KY) or

y¯Wmin(F(x),KY).

Then,

x¯ is a strict l-w minimum of SP1;

SF(x¯,y¯)(x)(-int(KY))=,forallxM.

Proof.

It is obvious that (b)(a)F(x)lF(x¯). Then if there exists a neighbourhood U of x¯ such that for each xUM the condition (a) holds, we deduce from Definition 10 that (i) holds. On the other hand if (a) holds we have (25)Fx-y¯-intKY=,xUM. 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 x¯ be a strict l-w minimum of SP1. If F(x)lF(x¯), W min(F(x¯))={y¯}, and F(x¯) has the KY-w minimal property, then (26)DFx¯,y¯x-intKY=,xM.

Corollary 16.

Let x¯ be an l-w minimum of SP1. Let y¯F(x¯). Let us suppose that there exists a neighbourhood U of x¯ such that for each xUM one of the following conditions is satisfied:

y¯F(x)+int(KY) or

y¯Wmin(F(x),KY).

Then

x¯ is a strict l-w minimum of SP1;

DFx¯,y¯x-intKY=,forallxM.

Remark 17.

If x¯ is a strict l-w minimum 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 Rodríguez-Marín : let F:(0,2)R2 be defined by (27)Fx=y,zR2y-2x+22+z2x2,ifx1,y,zR2y-2x+22+z22-x2,ifx>1. Let KY=R+2. Then x¯=1 is a strict l-w minimum of F and y¯=(-1,0)Wmin(F(1)). But observe that Wmin(F(x¯)){y¯} and for all neighbourhoods U of x¯ there exists xUM such y¯F(x)+int(KY) and y¯Wmin(F(x)). On the other hand for all k-int(KY) and xM we take tn=0 for each nN, then for every sequence (xn,yn)(x,k) we have (28)-1,0+tnynF1+tnxn,nN, that is (29)DF1,-1,0x-intKY,xM. Hence (30)SF1,-1,0x-intKY,xM.

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 (31)F,Gx=Fx,Gx. Let z¯G(x¯)(-KZ) and we consider the following problem SP3 with respect to KY×(KZ+z¯): SP3l-minimizeF,Gx,subjecttoxM.

The following result compares the set of strict l-w minimum solution of SP2 to the set of strict l-w minimum solution of SP3.

Proposition 18.

If x¯ is a strict l-w minimal solution of SP2 then for all z¯G(x¯)(-KZ), x¯ is a strict l-w minimal solution of SP3 with respect to KY×(KZ+z¯),

Proof.

Suppose the contrary; then, for every neighbourhood U of x¯, there exists xUM such that (32)F,Gx¯F,Gx+intKY×KZ+z¯, then, (33)Fx¯Fx+intKY,Gx¯Gx+intKZ+z¯,and since z¯G(x¯), we get (34)Fx¯Fx+intKY,0Gx+KZ. Thus for every neighbourhood U of x¯ there exists xUM such that (35)Fx<lFx¯,Gx-KZ. This contradicts x¯ is a strict l-w minimal solution of SP2.

Let us formulate necessary conditions for the problem SP2. In the sequel we consider the following subset of M: (36)S=xMGx-KZ.

Theorem 19.

Let x¯ be a strict l-w minimum solution of SP2 and z¯G(x¯)(-KZ). If F(x¯) has the KY-w minimal property and Wmin(F(x¯),KY)={y¯}, then (37)DF,Gx¯,y¯,z¯x-intKY×KZ+z¯=,xM.

Proof.

Suppose the contrary; then there exist xM and (y,z)Y×Z such that (38)y,zDF,Gx¯,y¯,z¯x-intKY×KZ+z¯,and hence there exist tnnN0+ and (xn,yn,zn)(x,y,z) such that(39)tnyn,znF,Gx¯+tnxn-y¯,z¯,nN.On the other hand, (y,z)-int(KY×(KZ+z¯)), then there exists n1N such that (40)tnyn-intKY,zn-intKZ+z¯,nn1, and hence (41)tnyn-intKY,z¯+tnzn-KZ+1-tnz¯,nn1, as tn0+ there exists n2N such that 1-tn>0 for every nn2, then (1-tn)z¯-KZ. Let N=max(n1,n2), by (39) we get(42)Fx¯+tnxn-y¯-intKY,Gx¯+tnxn-KZnN.From hypothesis we have x¯ is a strict l-w minimum solution of SP2; then there exists a neighbourhood U of x¯ such that FxlF(x¯) for all xUS. Since x¯+tnxnx¯ then there exists n0N such that x¯+tnxnUM for all nn0; thus x¯+tnxnUS for all nmax(n0,N); hence, (43)Fx¯+tnxnlFx¯nmaxn0,N, and, by Lemma 12 and hypothesis, we get (44)Fx¯+tnxn-y¯-intKY=nmaxn0,N. This contradicts (42).

As an immediate consequence we have the following corollary.

Corollary 20.

Let x¯ be an l-w minimum of SP2 and z¯G(x¯)(-KZ). Let y¯F(x¯). Let us suppose that there exists a neighbourhood U of x¯ such that for each xUS one of the following conditions is satisfied:

y¯F(x)+int(KY) or

y¯Wmin(F(x),KY).

Then

x¯ is a strict l-w minimum of SP2;

D(F,G)(x¯,y¯,z¯)(x)(-int(KY×(KZ+z¯)))=,forallxM.

4. Sufficient Optimality Conditions

It is well known from vector optimization that we can derive sufficient condition under Γ-contingently quasiconvex assumptions. Next, we establish sufficient condition with similar assumptions for set optimization. The following terminology is used. Let Y denote the dual space of Y, and let (45)KY+=uYuy0,yKY denote the nonnegative dual cone of KY.

The next theorem provides a sufficient condition for the l-w minimum solution of SP2.

Theorem 21.

Let (x¯,y¯,z¯)Gr(F,G) and M-x¯domDF,Gx¯,y¯,z¯. Assume that there are uKY+0Y and vKZ+ such that v(z¯)=0 and(46)uy+vz0,for  everyy,zDF+KY,G+KZx¯,y¯,z¯x-x¯xM.If F+KY,G+KZ:M^Y×Z is Γ-contingently quasiconvex at (x¯,y¯,z¯) with (47)M^=xMGx-KZ+conez¯-conez¯,Γ=-intKY×-KZ+conez¯-conez¯,then x¯ is an l-w minimal solution of SP2 on M^.

Proof.

Let us show that, for every xM^,(48)DF+KY,G+KZx¯,y¯,z¯x-x¯Γ=.Assume the contrary; then there exist xM^ and (y,z)Y×Z such that (49)y,zDF+KY,G+KZx¯,y¯,z¯x-x¯,y-intKY,z-KZ+conez¯-conez¯.Since uKY+0Y, vKZ+, and v(z¯)=0, we have (50)uy+vz<0. This contradicts (46).

On the other hand, we have that F+KY,G+KZ is Γ-contingently quasiconvex at (x¯,y¯,z¯); thus (48) ensures that there is no xM^ such that (51)Fx+KY,Gx+KZ-y¯,z¯Γ, that is (52)Fx+KY-y¯-intKY,Gx+KZ-z¯-KZ+conez¯-conez¯. Hence for every xM^ there exists y¯F(x¯) such that (53)Fx-y¯-intKY=, by Proposition 3, and we deduce that x¯ is a l-w minimal solution of SP2 on M^.

As an immediate consequence we have the following corollary.

Corollary 22.

Under the setting of Theorem 21, if the map (F,G):M^Y×Z is KY×KZ-pseudoconvex at (x¯,y¯,z¯) then x¯ is a l-w minimal solution of SP2 on M^.

From Theorem 21, we obtain the following sufficient optimality condition for SP1.

Theorem 23.

Let (x¯,y¯)Gr(F) and M-x¯domDFx¯,y¯. Assume that there exists uKY+0Y such that (54)uy0,for  every  yDF+KYx¯,y¯x-x¯xM. If F+KY:MY is Γ-contingently quasiconvex at (x¯,y¯) with, Γ=-int(KY), then x¯ is an l-w minimal solution of SP1.

The following two corollaries of the above result are immediate.

Corollary 24.

Let (x¯,y¯)Gr(F) and M-x¯dom[DF(x¯,y¯)]. Assume that (55)DF+KYx¯,y¯x-x¯-intKY=,xM. If F+KY:MY is Γ-contingently quasiconvex at (x¯,y¯) with, Γ=-int(KY), then x¯ is an l-w minimal solution of SP1.

Corollary 25.

Let (x¯,y¯)Gr(F) and M-x¯domDFx¯,y¯. Assume that (56)DF+KYx¯,y¯x-x¯-intKY=,xM. If F:MY is KY-pseudoconvex at (x¯,y¯), then x¯ is an l-w minimal solution of SP1.

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 S-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.

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