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.

Vector variational inequality (VVI) was first introduced by Giannessi [

Note that some results regarding gap functions for VVLI with set-valued mappings have appeared in the literature [

Note also that there are some papers discussing solution relationships between set-valued optimization problems and vector variational-like inequalities. Miholca [

Inspired and motivated by the works [

Let

Let

In the following sections, we denote

Let

Let

Recall that the contingent cone

Suppose that

a continuous linear map

the set

The following notions are based on the concept of contingent epiderivative.

Let

Let assumptions of Lemma

For the spacial case, where

If

We propose the following relaxed compactness which will be needed for the following sections.

(i) Let

(ii) Let

sequence

sequence

(iii) A subset

(iv) If, in (iii), pointwise convergence, that is,

(i) If

(ii) If

(iii) If

By Remark

From Remark

Let

A set

Obviously, the convex set is a particular case of the invex set if

Let

Throughout this paper, we always assume that

(SGVVLI) find

(WGVVLI) find

In this section, we assume that

Let

Let

Let

Let

Suppose that

The function

The function

(i) By assumptions, for each

(ii) Since for any given

Let

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 [

We denote set-valued optimization problems (

A pair

The set of all weak efficient solutions of (SOP) is denoted by

Let

We note that

Let

Let a multifunction

The following lemma will give a similar but more generalized result.

Let a multifunction

Take a system

Let

Taking arbitrary elements

Let

Let

Let

Let

Suppose that

Let

The following form of

Let

for all

if there is a nonempty subset

(i) Let

(ii) Assume that there exists a nonempty compact convex set

Then, VOP has a generalized weakly efficient solution.

We define a multivalued map

We note that our assumptions in Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Science Foundation of China (nos. 61373174 and 11301407).