# Generalized Stampacchia Vector Variational-Like Inequalities and Vector Optimization Problems Involving Set-Valued Maps

Authors: Yanfei Chai; Sanyang Liu; Guotao Wang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We first obtain that subdifferentials of set-valued mapping from finite-dimensional spaces to finite-dimensional  possess certain relaxed compactness. Then using this weak compactness, we establish gap functions for generalized Stampacchia vector variational-like inequalities which are defined by means of subdifferentials. Finally, an existence result of generalized weakly efficient solutions for vector optimization problem involving a subdifferentiable and preinvex set-valued mapping is established by exploiting the existence of a solution for the weak formulation of the generalized Stampacchia vector variational-like inequality via a Fan-KKM lemma.


---

## Body

## 1. Introduction

Vector variational inequality (VVI) was first introduced by Giannessi [1] in finite-dimensional spaces. Since then, much attention has been given to investigate VVIs and their generalizations. For details on VVI and their generalizations, we refer to [2–4] and the references therein. The vector variational-like inequality (VVLI), a generalization of VVI, was studied in [5–9] with applications in vector optimization problem (VOP). In [7], the equivalences among Minty VVLI, Stampacchia VVLI, both for η-subdifferentiable functions (see [10]) and for nondifferentiable nonconvex VOP were established and an existence theorem for the so-called generalized weakly efficient solutions of nondifferentiable nonconvex VOP was obtained through the relationship between VVLI and VOP. The VVLI approach was also used in [9, 11] to prove some existence theorems of generalized efficient solutions for nondifferentiable invex VOP. Particularly, VVLI with set-valued mapping was also considered in [12–14].Note that some results regarding gap functions for VVLI with set-valued mappings have appeared in the literature [12, 14, 15], but so far to the best of our knowledge there is no result available in the literature about a gap function for a generalized Stampacchia vector variational-like inequality (GVVLI) which is defined by means of generalized subdifferentials of set-valued mapping (see Definition 3). In this paper, we introduce gap functions for our GVVLI.Note also that there are some papers discussing solution relationships between set-valued optimization problems and vector variational-like inequalities. Miholca [13] and Zeng and Li [16] considered several kinds of generalized invexity for set-valued mappings and established some solution relationships between set-valued optimization problems and generalized vector variational-like inequalities problems ((MVVI) and (VVLIS) in [13, 16], resp.), but they did not consider the existence results for their solutions.Inspired and motivated by the works [7, 12–16], in this paper, we study GVVLI. We first establish generalized subdi erential mapping ∂F of set-valued mapping F from finite-dimensional space into finite-dimension is an asymptotically compact, closed and convex-valued mapping, and then, we introduce gap functions for our GVVLI problems. In the final section of this paper, we discuss solution relationships between GVVLI and vector optimization problem involving set-valued mapping (VOP); then we obtain existence of solutions for GVVLI by using a Fan-KKM lemma and consequently an existence result for weak efficient solution of VOP is established.


## 2. Preliminaries and Notations

LetRn be the n dimensional space, where n is a given positive integer. Set
(1)R+n={y=(y1,y2,…,yn)T∈Rn∣yi≥0,i=1,2,…,n},int⁡R+n={y=(y1,y2,…,yn)T∈Rn∣yi>0,i=1,2,…,n},
where the symbol T denotes the transpose. A nonempty subset D of Rn is said to be a cone if λD⊆D for all λ≥0; D is said to be a convex cone if D is a cone and D+D⊆D; D is a closed cone if D is a cone and closed; and D is called a pointed cone if D is a cone and D∩(-D)={0}. Clearly, R+n is a closed, convex, and pointed cone of Rn, and int R+n is a convex pointed cone of Rn. We consider the orderings induced by R+n and int R+n in the following form.Lety1,y2∈Rn:
(2)y1≤y2⟺y2-y1∈R+n,y1≰y2⟺y2-y1∉R+n,y1<y2⟺y2-y1∈int⁡R+n,y1≮y2⟺y2-y1∉int⁡R+n.In the following sections, we denoteD=R+n. Let Rm be the m-dimensional space. Denote by L(Rm,Rn) the space of all the continuous linear mappings from Rm to Rn and by 〈l,x〉 the value of l∈L(Rm,Rn) at x∈Rm. Let A⊂Rm. For x∈Rm, denote by d(x,A) the distance from x to A; that is, d(x,A)=inf⁡a∈A∥x-a∥. Let F:X→Y be a set-valued mapping. The graph, the epigraph, and the domain of F are defined, respectively, by
(3)grF={(x,y)∈X×Y:y∈F(x)},epi⁡F={(x,y)∈X×Y:y∈F(x)+D},dom⁡F={x∈X:F(x)≠∅}.Definition 1 (see [17, 18]).
LetA⊂X; the contingent cone of A at x0 is defined by
(4)T(A,x0)={u∈X:∃tn⟶0+,∃un⟶u,∃n0∈N,fiii∀n≥n0,x0+tnun∈A}.Definition 2 (see [17, 18]).
Letx0∈A and let a pair (x0,y0)∈grF be given. The contingent epiderivative DF(x0,y0) of F at (x0,y0) is the single-valued mapping from X to Y defined by
(5)epi⁡(DF(x0,y0))=T(epi⁡F,(x0,y0)).F is called epidifferentiable at (x0,y0) if the contingent epiderivative DF(x0,y0) of F at (x0,y0) exists.Recall that the contingent coneT(epi⁡F,(x0,y0)) consists of all tangent vectors h=lim⁡n→+∞λn((xn,yn)-(x0,y0)) with (x0,y0)=lim⁡n→+∞(xn,yn) and (xn,yn)∈epi⁡F for all n∈N and λn>0.Definition 3 (see [19]).
Suppose thatF is epidifferentiable at (x0,y0)∈grF; then(i)
a continuous linear mapl:X→Y, with l(x)≤DF(x0,y0)(x), for all x∈dom⁡(DF(x0,y0)) is called a subgradient of F at (x0,y0);(ii)
the set∂F(x0,y0)={l:X→Y:l(x)≤DF(x0,y0)(x),  ∀x∈dom⁡(DF(x0,y0)} of all subgradients l of F at (x0,y0) is called the subdifferential of F at (x0,y0).The following notions are based on the concept of contingent epiderivative.Lemma 4 (see [19]).
LetX and Y be real normed spaces, A a subset of X, and D a convex cone. F:A→2Y, (x0,y0)∈grF, and let contingent epiderivative DF(x0,y0) of F at (x0,y0) exist. Then the subdifferential ∂F(x0,y0) is convex.Lemma 5 (see [19]).
Let assumptions of Lemma4 be satisfied and let D be closed. If all subgradients are bounded, then the subdifferential is closed in the linear space of all bounded maps.For the spacial case, whereX=Rm and Y=Rn, linear maps are always bounded, so the subdifferential is closed whenever D is closed.Corollary 6.
IfX=Rm, Y=Rn, and D=R+n, then subdifferential of F at (x,y)∈grF is a closed and convex set in the linear space L(Rm,Rn).We propose the following relaxed compactness which will be needed for the following sections.Definition 7 (see [20]).
(i) Letln and l be in L(X,Y). The sequence {ln} is said to pointwisely converge to l and written as l=p-lim⁡ln if lim⁡n→+∞ln(x)=l(x) for all x∈X.
(ii) Letln and l be in L(X,Y). The sequence {ln} is said to asymptotically pointwisely converge to l if(a)
sequence{ln} is bounded and it has a subsequence {lnk} and l∈L(X,Y) such that l=p-lim⁡lnk;(b)
sequence{ln} with lim⁡∥ln∥=∞ and the sequence {ln/∥ln∥} has a subsequence which pointwisely converges to some l∈L(X,Y)∖{0}.
(iii) A subsetM⊂L(X,Y) is called asymptotically pointwisely compact or asymptotically p-compact if each sequence {ln}⊂M has a subsequence which asymptotically pointwisely converges to l∈L(X,Y).
(iv) If, in (iii), pointwise convergence, that is,p-lim, is replaced by convergence, that is, lim, a subset M⊂L(X,Y) is called asymptotically compact.Remark 8.
(i) IfX and Y are finite dimensional, a convergence occurs if and only if the corresponding pointwise convergence does.
(ii) IfX and Y are finite dimensional, every subset is asymptotically p-compact and asymptotically compact.
(iii) IfX and Y are finite dimensional, then every subset of L(X,Y) is asymptotically p-compact.Remark 9.
By Remark8, if assumptions of Corollary 6 are satisfied and ∂F(x,y) exists at (x,y)∈grF, then ∂F(x,y) is an asymptotically compact closed and convex subset in L(Rm,Rn).Remark 10.
From Remark9, if t∂F(x,y)⊆∂F(x,y) for all t∈(0,1], whenever ∂F(x,y) exists at (x,y)∈grF, then each sequence {ln}⊂∂F(x,y) has a subsequence which asymptotically converges to some l∈∂F(x,y). In the following sections we always assume that t∂F(x,y)⊆∂F(x,y) for all t∈(0,1], whenever ∂F(x,y) exists at (x,y)∈grF.Example 11.
LetX=Y=R, A=[0,3], and D=R+={t∈R:t≥0} and let F be defined by
(6)F(x)={y∈R∣x≤y≤2x+1}.
Then (0,0)∈grF and DF(0,0)(x)=x for all x≥0(dom⁡(DF(0,0))=[0,+∞)); therefore, ∂F(0,0)={l∈R∣-∞<l≤1}. Obviously, t∂F(0,0)⊆∂F(0,0) for all t∈(0,1].Definition 12.
A setA⊂X is said to be an invex set if there exists a function η:X×X→X such that ∀x,y∈A, ∀λ∈[0,1], and y+λη(x,y)∈A.Remark 13.
Obviously, the convex set is a particular case of the invex set ifη(x,y)=x-y, but the converse does not hold; see the following example.Example 14.
LetA=[-3,0]∪[1,2]⊂R and let η be defined by
(7)η(x,y)={x-y,x,y∈[-3,0]orx,y∈[1,2],x,x∈[-3,0],y∈[1,2],0,x∈[1,2],y∈[-3,0].
We note that A is an invex set with respect to the above η.Throughout this paper, we always assume thatA is an invex subset of Rm, the function η is defined on A, that is, η:A×A→Rm, and F:A→2Rn is a set-valued mapping with ∂F(x,y) existing at every (x,y)∈grF and x∈A. By Remark 9, we obtain ∂F:Rm×Rn→2L(Rm,Rn), when it exists, a set-valued mapping with an asymptotically compact closed convex-valued ∂F(x,y) at (x,y)∈grF. Furthermore, under mild assumptions, ∂F(x,y) has certain compactness (see Remark 10). Next, we consider the following strong and weak generalized Stampacchia vector variational-like inequalities (GVVLI):(SGVVLI) find(x0,y0)∈grF and l0∈∂F(x0,y0) such that
(8)〈l0,η(x,x0)〉≮int⁡D0,∀x∈Aand(WGVVLI) find(x0,y0)∈grF such that ∀x∈A and there exist l0(x)∈∂F(x0,y0) satisfying
(9)〈l0(x),η(x,x0)〉≮int⁡D0.We denote by SSGVVLI and SWGVVLI the solution sets of SGVVLI and WGVVLI, respectively. Obviously, SSGVVLI⊆SWGVVLI.


## 3. Gap Functions

In this section, we assume thatη defined on A is an open map such that η(x,z)+η(z,x)=0 for all x,z∈A. Then, we introduce the concept of gap functions for our SGVVLI and WGVVLI.Definition 15.
LetA be the domain of SGVVLI (resp., WGVVLI). A function h:A×Rn→R∪{+∞} is said to be a gap function for SGVVLI (resp., WGVVLI) if it satisfies the following properties:(i)
h(x,y)≥0, ∀x∈A with y∈F(x);(ii)
h(x0,y0)=0 if and only if (x0,y0) (where y0∈F(x0)) solves SGVVLI (resp., WGVVLI).Letx∈A with y∈F(x), u∈dom⁡(DF(x,y)), and l∈∂F(x,y). Denote
(10)〈l,u〉=(〈l,u〉1,…,〈l,u〉n);
that is, 〈l,u〉i is the ith component of 〈l,u〉, i=1,2,…,n. Now, we define φ∂F:A→R∪{+∞} as
(11)φ∂F(x,y)=inf⁡l∈∂F(x,y)sup⁡z∈Amin⁡1≤i≤n〈l,η(x,z)〉i,∀x∈Awithy∈F(x),
where y∈F(x) such that ∂F(x,y) exists. For x∈A and y∈F(x), let
(12)Sxy={l∣l:A→∂F(x,y)},
where  Sxy is the set of all operators l from A to ∂F(x,y). Let x∈A, (x,y)∈grF, and l∈Sxy; then l(z)∈∂F(x,y), ∀z∈A. We also define ϕ∂F:A→R∪{+∞} as follows:
(13)ϕ∂F(x,y)=inf⁡l∈Sxysup⁡z∈Amin⁡1≤i≤n〈l(z),η(x,z)〉i,∀x∈Awithy∈F(x).Lemma 16.
LetX be a normed space and ∅≠M⊆X an asymptotically compact closed convex subset; then there exists x0∈M such that ∥x0∥=inf⁡{∥x∥∣x∈M}.Proof.
Letd=inf⁡x∈M{∥x∥}; then for any n∈N, there exists xn∈M such that
(14)d≤∥xn∥<d+1n≤d+1.
So sequence {xn} is bounded; by assumption M being an asymptotically compact closed convex subset, there exists a subsequence {xnk} of {xn} such that xnk→x0∈M. For x0, by applying Hahn-Banach theory, there exists f∈X* such that ∥f∥=1, f(x0)=∥x0∥. Hence, on the one hand,
(15)x0∈M⟹∥x0∥≥d;
on the other hand,
(16)∥x0∥=f(x0)=lim⁡k→∞f(xnk)≤lim⁡_k→∞⁡∥f∥∥xnk∥=d.
Therefore, ∥x0∥=d=inf⁡{∥x∥∣x∈M}.Theorem 17.
Suppose that∂F(x,y)≠∅ for all x∈A with y∈F(x) such that t∂F(x,y)⊆∂F(x,y) for all t∈(0,1]. If the function η is an open map such that η(x,z)+η(z,x)=0 for all x,z∈A. Then, the following statements are true.(i)
The functionφ∂F defined by (11) is a gap function for SGVVLI.(ii)
The functionϕ∂F defined by (13) is a gap function for WGVVLI.Proof.
(i) By assumptions, for eachx∈A with (x,y)∈grF, ∂F(x,y) is an asymptotically compact closed convex set. Then, the function φ∂F given by (11) is well defined and from Lemma 16 we have
(17)φ∂F(x,y)=min⁡l∈∂F(x,y)sup⁡z∈Amin⁡1≤i≤n〈l,η(x,z)〉i,∀x∈Awith(x,y)∈grF.
It is immediate that
(18)sup⁡z∈Amin⁡1≤i≤n〈l,η(x,z)〉i≥0,∀x∈Awith(x,y)∈grF,∀l∈∂F(x,y),
so that φ∂F(x,y)≥0, ∀x∈A with y∈F(x). Let x0∈A with y0∈F(x0). We note that φ∂F(x0,y0)=0 if and only if there exists l0∈∂F(x0,y0) such that
(19)sup⁡z∈Amin⁡1≤i≤n〈l0,η(x0,z)〉i=0,
or
(20)min⁡1≤i≤n〈l0,η(x0,z)〉i=0,∀z∈A,
which is equivalent to
(21)〈l0,η(z,x0)〉≰int⁡D0,∀z∈A;
that is, (x0,y0)∈SSGVVLI.
(ii) Since for any givenx∈A with y∈F(x) and l∈Sxy(22)sup⁡z∈Amin⁡1≤i≤n〈l(z),η(x,z)〉i≥min⁡1≤i≤n〈l(x),η(x,x)〉i=0,
we obtain
(23)ϕ∂F(x,y)=inf⁡l∈Sxysup⁡z∈Amin⁡1≤i≤n〈l(z),η(x,z)〉i≥0,∀x∈Awithy∈F(x).
We assume that (x0,y0)∈grF solves WGVVLI. Then, for any z∈A, there is a l(z)∈∂F(x0,y0) such that
(24)〈l(z),η(z,x0)〉≰int⁡D0,
which implies that
(25)min⁡1≤i≤n〈l(z),η(x0,z)〉i≤0.
Thus an operator l from A to ∂F(x0,y0) has been defined. It follows that l∈Sx0y0 and
(26)min⁡1≤i≤n〈l(z),η(x0,z)〉i≤0,∀z∈A.
Therefore,
(27)sup⁡z∈Amin⁡1≤i≤n〈l(z),η(x0,z)〉i≤0,
and so
(28)ϕ∂F(x0,y0)=inf⁡l∈Sx0y0sup⁡z∈Amin⁡1≤i≤n〈l(z),η(x0,z)〉i≤0.
Since ϕ∂F(x,y)≥0 for all x∈A with y∈F(x), it follows that ϕ∂F(x0)=0. Conversely, suppose that ϕ∂F(x0)=0 and y0∈F(x0). Let
(29)sup⁡z∈Amin⁡1≤i≤n〈l(z),η(x0,z)〉i,∀l∈Sx0y0.
Since ϕ∂F(x0)=0, for ϵ1>ϵ2>⋯>ϵm>⋯>0 with ϵm=ϵ1/2m-1, there exist lm∈Sx0y0 such that
(30)sup⁡z∈Amin⁡1≤i≤n〈lm(z),η(x0,z)〉i≤ϵm,
or equivalently
(31)min⁡1≤i≤n〈lm(z),η(x0,z)〉i≤ϵm,∀z∈A,∀m∈N,
which implies
(32)min⁡1≤i≤n〈lm(z)∥lm(z)∥,η(x0,z)〉i≤ϵm∥lm(z)∥,∀z∈A,∀m∈N.
Observe that {lm(z)}⊆∂F(x0,y0) for all z∈A. Since ∂F(x0,y0) is asymptotically compact closed convex set and by Remark 10, for any z∈A, {lm(z)} has an asymptotically convergent subnet with limit l0(z)∈∂F(x0,y0). Without loss of generality, we may assume that {lm(z)} asymptotically converges to l0(z)∈∂F(x0,y0). Consequently, an operator l0:A→∂F(x0,y0) has been defined; that is, l0∈Sx0y0 and
(33)〈lm(z),η(x0,z)〉⟶〈l0(z),η(x0,z)〉,∀z∈A,
or
(34)〈lm(z)∥lm(z)∥,η(x0,z)〉⟶〈l0(z),η(x0,z)〉,∀z∈A,
which imply
(35)min⁡1≤i≤n〈lm(z),η(x0,z)〉i⟶min⁡1≤i≤n〈l0(z),η(x0,z)〉i,∀z∈A,
or
(36)min⁡1≤i≤n〈lm(z)∥lm(z)∥,η(x0,z)〉i⟶min⁡1≤i≤n〈l0(z),η(x0,z)〉i,∀z∈A.
Taking the limit for m→+∞ in (31) or (32), we obtain
(37)min⁡1≤i≤n〈l0(z),η(x0,z)〉i≤0,∀z∈A.
Then, for any z∈A, there exists l0(z)∈∂F(x0,y0) such that
(38)min⁡1≤i≤n〈l0(z),η(z,x0)〉i≥0,
which implies that, for any z∈A, there exists l0(z)∈∂F(x0,y0) such that
(39)〈l0(z),η(z,x0)〉≰int⁡D0;
that is, (x0,y0)∈SWGVVLI. This completes the proof.Example 18.
LetX=Y=R, A=[0,5], and D=R+={t∈R:t≥0} and let F and η be defined by
(40)F(x)={y∈R∣0≤y≤-x+5},η(x,y)=12(x-y),∀x,y∈A.
Then, η(x,y)+η(y,x)=0 for all x,y∈A. Since (0,0)∈grF and
(41)T(epi⁡F,(0,0))={(x,y)∣x≥0,y≥0},
therefore, we get
(42)DF(0,0)(x)={0},
when x∈dom⁡(DF(0,0)), and, consequently,
(43)∂F(0,0)={l∈R∣-∞<l≤0}.
So, t∂F(0,0)⊆∂F(0,0) hold for all t∈(0,1].


## 4. Existence Result for Vector Optimization Problems

Vector variational inequalities (or their generalized form) have been shown to be a useful tool in vector optimization. Some authors have proved the equivalence between them; see [8, 13, 16]. In this section, we prove the existence of generalized weakly efficient solutions for the following invex vector optimization problem (VOP) through the relationship between GVVLI and VOP by using a Fan-KKM lemma:
(44)Min⁡{F(x):x∈A},
where F is a set-valued mapping as above and A is an invex set.We denote set-valued optimization problems (44) as (SOP). Let x0∈A, y0∈F(x0).Definition 19.
A pair(x0,y0)∈grF is called weak efficient solution of F on A if (F(A)-y0)⋂‍(-int⁡D)=∅.The set of all weak efficient solutions of (SOP) is denoted byWMin⁡(F,A).Definition 20.
LetF:A→2Y be a set-valued mapping, x0∈A, and (x0,y0)∈grF. F is called strict pseudoinvex with respect to η:A×A→X at (x0,y0) if ∀x∈A, ∀y∈F(x), and ∀l∈∂F(x0,y0), y-y0∉int⁡D⇒lη(x,x0)∉int⁡D.Remark 21.
We note thatF of Example 18 is strict pseudoinvex with respect to η at (0,0). In fact, since 0∈F(x) for all x∈A and taking any l∈∂F(0,0), we have 0-0=0∉int⁡R+ and obviously lη(u,0)=(1/2)l(u-0)=(1/2)l(u)≤0∉int⁡R+, where u∈dom⁡(DF(0,0)) and u≥0.Definition 22.
LetF:A→2Y be a set-valued mapping. F is said to be D-preinvex with respect to η on A if ∀x,z∈A, ∀λ∈[0,1], λF(x)+(1-λ)F(z)⊂F(z+λη(x,z))+D.Definition 23.
F is said to be upper semicontinuous at x-∈A if for any neighbourhood V of the origin in Y there exists a neighbourhood U of the origin in X such that
(45)F(x)⊆F(x-)+V,∀x∈(x-+U)∩A.Lemma 24 (Lemma 1.1, [21]).
Let a multifunctionF:X→2Y. If F is compact-valued multifunction, then F is upper semicontinuous if and only if for every net {(xn,yn)} in grF that satisfies xn→x for some x∈X the net {yn} has a subnet converging to a point in F(x).The following lemma will give a similar but more generalized result.Lemma 25.
Let a multifunctionF:X→2Y. If F is asymptotically closed compact-valued multifunction and upper semicontinuous such that tF(x)⊆F(x) for all t∈(0,1] and x∈X, then for every net {(xn,yn)} in grF that satisfies xn→x for some x∈X the net {yn} has a subnet asymptotically converging to a point in F(x).Proof.
Take a system{Vμ}, μ∈I of neighbourhoods of the origin in Y such that {Vμ}→{0}, where {Vμ}→{0} stands for Vν⊆Vμ, ν≥μ, and ⋂μ∈I‍Vμ={0} and I is an ordered index set. From the upper semicontinuity of F and xn→x, one claims that for any Vμ there exists n0=n0(μ) such that F(xn)⊆F(x)+Vμ for all n≥n0. Consequently, yn∈F(xn) can be written as yn=y¯n+vμ with y¯n∈F(x) and vμ∈Vμ. From the asymptotic compactness of F(x) and {Vμ}→{0}, without loss of generality, we may assume that {y¯n} asymptotically converges to y∈Y. Furthermore, since tF(x)⊆F(x) for all t∈(0,1] and the closedness assumption of F(x), we can conclude that y∈F(x). This implies that {yn} asymptotically converges to y∈F(x). The proof is complete.Proposition 26.
LetA be an invex set with respect to η and let F:A→2Y be D-preinvex with respect to the same η on A. Let x0∈A, (x0,y0)∈grF, and ∂F(x0,y0)≠∅. Then ∀l∈∂F(x0,y0), ∀x∈A, y∈F(x), and
(46)y-y0∈〈l,η(x,x0)〉+D.Proof.
Taking arbitrary elementsx∈A and y∈F(x), for all n∈N, we define a sequence (xn,yn)n∈N with
(47)xn=x0+1nη(x,x0),yn=y0+1n(y-y0).
Since A is an invex set and F is a D-preinvex map, it follows for all n∈N(48)xn=x0+1nη(x,x0)∈A,yn=(1-1n)y0+1ny∈F(x0+1nη(x,x0))+D=F(xn)+D.
So, (xn,yn)n∈N is a sequence in the epigraph of F converging to (x0,y0). Moreover we obtain
(49)lim⁡n→∞n(xn-x0,yn-y0)=(η(x,x0),y-y0).
Consequently, we get
(50)(η(x,x0),y-y0)∈T(epi⁡F,(x0,y0))=epi⁡(DF(x0,y0)),
implying that
(51)y-y0∈DF(x0,y0)(η(x,x0))+D.
It is clear that
(52)∀x∈A,η(x,x0)∈dom⁡DF(x0,y0).
By the definition of the subdifferential ∂F(x0,y0), for all l∈∂F(x0,y0), we have
(53)〈l,η(x,x0)〉≤DF(x0,y0)(η(x,x0)),∀η(x,x0)∈dom⁡DF(x0,y0);
that is,
(54)DF(x0,y0)η(x,x0)∈〈l,η(x,x0)〉+D.
Considering (51) and (54), we conclude that y-y0∈〈l,η(x,x0)〉+D.Proposition 27.
LetA be a nonempty invex subset of Rm and let η:A×A→Rm be an open mapping such that it is affine in the first argument and η(x,z)+η(z,x)=0, ∀x,z∈A. Let F:A→2Rn be strict pseudoinvex with respect to the same η and ∂F(x,y)≠∅ for all (x,y) with y∈F(x) such that t∂F(x,y)⊆∂F(x,y) for all t∈(0,1]. If (x0,y0)∈WMin⁡(F,A) and ∂F is upper semicontinuous at (x0,y0), then (x0,y0)∈SWGVVLI.Proof.
Let(x0,y0)∈WMin⁡(F,A). Consider any x∈A and any sequence {λn}→0 with λn∈(0,1]. Since A is invex,
(55)xn=x0+λnη(x,x0)∈A.
Since (x0,y0)∈WMin⁡(F,A), we have
(56)(F(xn)-y0)∩(-int⁡D)=∅.
That is,
(57)y0-yn∉int⁡D,∀yn∈F(xn).
Since DF(x0,y0) exists and η(x,x0)∈dom⁡DF(x0,y0), there exists yn∈F(xn) such that yn→y0 as λn→0+. Moreover, ∂F(x,y)≠∅ at every (x,y) with y∈F(x) and F is strict pseudoinvex with respect to η, so for all ln∈∂F(xn,yn) we have
(58)〈ln,η(x0,xn)〉∉int⁡D.
Since η is affine in the first argument and η(x,z)+η(z,x)=0, ∀x,z∈A, we have
(59)〈ln,η(x,x0)〉∉-int⁡D;
that is,
(60)〈ln,η(x,x0)〉∈Y∖-int⁡D,
where Y∖-int⁡D is a closed cone. By assumptions and Remark 9, we get that ∂F(xn,yn) is asymptotically compact closed convex set in L(Rn,Rm). Furthermore, ∂F being upper semicontinuous at (x0,y0), (xn,yn)→(x0,y0) and by Remark 10 and Lemma 25, consequently, {ln} asymptotically converges to some l0∈∂F(x0,y0) with ln∈∂F(xn,yn). If sequence {ln} is bounded, without loss of generality, we may assume that ln→l0∈∂F(x0,y0); otherwise, if sequence {ln} with lim⁡∥ln∥=∞, we may also assume sequence ln/∥ln∥→l0∈∂F(x0,y0); from these facts and Y∖-int⁡D being a closed cone, it follows that
(61)〈l0,η(x,x0)〉∈Y∖-int⁡D.
Thus, ∀x∈A, ∃l0∈∂F(x0,y0) such that
(62)〈l0,η(x,x0)〉∉-int⁡D.
Hence, (x0,y0)∈SWGVVLI.Remark 28.
LetA=[0,5] and η(x,y)=α(x-y); then A is an invex set and η is affine in the first argument and continuous in the second argument, where 0<α<1.Proposition 29.
LetA be a nonempty invex set of Rm and F:A→Rn subdifferentiable and D-preinvex with respect to the same η:A×A→Rm. If (x0,y0) is a solution of WGVVLI, then it is also a weakly efficient solution of VOP.Proof.
Suppose that(x0,y0) is a solution of WGVVLI, but not a generalized weakly efficient solution of VOP. Then there exist x-∈A such that F(x-)-y0∩-int⁡D≠∅. That is, ∃y-∈F(x-) such that y--y0∈-int⁡D. Since F is subdifferentiable and D-preinvex with respect to the same η, by Proposition 26, for each l0∈∂F(x0,y0), we have 〈l0,η(x-,x0)〉∈y--y0-D∈-int⁡D-D⊂-int⁡D, which contradicts (x0,y0)∈SWGVVLI.LetA be a nonempty convex subset of a vector space X. A mapping F:A→2X is said to be a KKM mapping if for each nonempty finite subset K of A, convK⊂F(K), where convK denotes the convex hull of K, and F(K)=⋃‍{F(x):x∈K}.The following form ofFan-KKM lemma appeared in [22].Lemma 30.
LetM be a convex subset of a Hausdorff topological vector space X, ∅≠A⊂M. Let Γ be set-valued maps such that the following conditions hold:(A1)
Γ is a KKM mapping on A;(A2)
for allx∈A, Γ(x) is closed;(A3)
if there is a nonempty subsetA0 of A such that the intersection ⋂x∈A0Γ(x) is compact and A0 is contained in a compact convex subset of M,then
(63)⋂x∈A‍Γ(x)≠∅.Theorem 31.
(i) LetA be a nonempty convex subset of Rm and let η:A×A→Rm be an open mapping such that it is affine in the first argument and continuous in the second argument and η(x,z)+η(z,x)=0, ∀x,  z∈A. Let F:A→2Rn be D-preinvex with respect to η such that ∂F(x,y) exists and ∂F is upper semicontinuous at every (x,y)∈grF with x∈A and that t∂F(x,y)⊆∂F(x,y) for all t∈(0,1]. If ∂F is upper semicontinuous at each (x,y)∈grF with x∈A.
(ii) Assume that there exists a nonempty compact convex setB⊂A such that for each z∈A∖B, ∃x~∈B, such that 〈∂F(z,w),η(x~,z)〉⊂-int⁡D, where w∈F(z).
Then, VOP has a generalized weakly efficient solution.Proof.
We define a multivalued mapΓ:A→2Rn by
(64)Γ(x)={z∈A:〈∂F(z,w),η(x,z)〉⊈-int⁡D}={z∈A:∃lz∈∂F(z,w)fffsuchthat〈lz,η(x,z)〉∈Y∖-int⁡D}ffffffffffffffffffffffffffffffffffffffffffffff∀x∈A,
where w∈F(z) such that ∂F(z,w)≠∅. Then, ∀x∈A, Γ(x) is closed in Rm. Indeed, let {zn} be a sequence in Γ(x) such that zn→z0∈A. Let w0∈F(z0) such that DF(z0,w0) exists. F is epidifferentiable at (z0,w0), so for each zn there exists wn∈F(zn) such that wn→w0. Furthermore, zn∈Γ(x) implies there exist lzn∈∂F(zn,wn) such that
(65)〈lzn,η(x,zn)〉∈Y∖-int⁡D.
Since ∂F is upper semicontinuous at (z0,w0), (zn,wn)→(z0,w0) and considering also Remark 10, {lzn} possesses an asymptotically convergent subnet. Without loss of generality, we may assume that the lzn→lz0 (or lzn/∥lzn∥→lz0) and lz0∈∂F(z0,w0). Since η is continuous in the second variable and Y∖-int⁡D is a closed set, it is easy to check that 〈lz0,η(x,z0)〉∈Y∖-int⁡D. So, Γ(x) is a closed set. Next, we claim that Γ is a KKM mapping on A. Suppose that Γ is not a KKM mapping; then there exist {x1,x2,…,xn}⊂A and ti≥0 with ∑i=1n‍ti=1 such that z=∑i=1n‍tixi∉⋃i=1n‍Γ(xi). Thus, ∃lz∈∂F(z,w) such that, for any i=1,2,…,n,
(66)〈lz,η(xi,z)〉∈-int⁡D.
Since int⁡D is a convex cone, we have
(67)〈lz,t1η(x1,z)〉+〈lz,t2η(x2,z)〉+⋯+〈lz,tnη(xn,z)〉∈-int⁡D.
Since η(·,z) is affine, we get
(68)〈lz,η(z,z)〉∈-int⁡D.
But from η(x,z)+η(z,x)=0, ∀x,z∈A, we have η(z,z)=0. Thus 〈lz,η(z,z)〉=0∈-int⁡D, which contradicts the fact 0∉-int⁡D, so Γ is a KKM mapping. By condition (ii), ⋂x∈BΓ(x) is a closed subset of a compact set B and hence compact. Then by Lemma 30, ⋂x∈AΓ(x)≠∅. That is, ∃x0∈A, y0∈F(x0) such that l0∈∂F(x0,y0) and 〈l0,η(x,x0)〉∉-int⁡D, ∀x∈A. Thus, (x0,y0)∈SWGVVLI. From Proposition 29, we obtain (x0,y0)∈WMin⁡(F,A). The proof is complete.Remark 32.
We note that our assumptions in Theorem31 are totally different from those of Theorem  8 in [5] and Theorem  4.5 in [9] since we assume that ∂F is upper semicontinuous and t∂F(x,y)⊆∂F(x,y) for all t∈(0,1] while correspondences T and ∂f in [5] and [9], respectively, were supposed to be pseudomonotone and Γ(x) needs to be closed.



---

*Source: 948270-2014-08-27.xml*