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The concept of sharp efficient solution for vector equilibrium problems on Banach spaces is proposed. Moreover, the Fermat rules for local efficient solutions of vector equilibrium problems are extended to the sharp efficient solutions by means of the Clarke generalized differentiation and the normal cone. As applications, some necessary optimality conditions and sufficient optimality conditions for local sharp efficient solutions of a vector optimization problem with an abstract constraint and a vector variational inequality are obtained, respectively.

Let

A vector

(i) Note that it always holds

(ii) If

(iii) Every local sharp efficient solution must be a local efficient solution for (VEP).

As we know, vector equilibrium problems cover various classes of optimization-related problems and models arisen in practical applications, such as vector variational inequalities, vector optimization problems, vector Nash equilibrium problems, and vector complementarity problem, see [

Recently, there has been increasing interest in dealing with optimality conditions for nonsmooth optimization problems by virtue of modern variational analysis techniques. Gong [

In this paper, by virtue of the Clarke generalized differentiation and the normal cone, we first establish a necessary optimality condition for the local sharp efficient solution of (VEP) without any convexity assumptions. And then, we obtain the sufficient optimality condition for the local sharp efficient solution of (VEP) under some appropriate convexity assumptions. Simultaneously, we show that the local sharp efficient solution and the sharp efficient solution are equivalent for the convex case. Finally, we apply our results, respectively, to get some necessary optimality conditions and sufficient optimality conditions for local sharp efficient solutions of a vector optimization problem with an abstract constraint and a vector variational inequality.

Throughout this paper, we denote by

The main tools for our study in the paper are the Clarke generalized differentiation notions which are generally used in variational analysis and nonsmooth analysis. We refer to [

Next, we collect some useful and important propositions for this paper.

For every nonempty closed subset

The following necessary optimality condition, called generalized Fermat rule, for a function to attain its local minimum is useful for our analysis.

Let

We recall the following sum rule for the Clarke subdifferential which is important in the sequel.

Let

The following chain rule of Clarke subdifferential is useful in the paper.

Let

Given a point

Given a point

If

Let

implies

(i) Since

(ii) Since

Assume that

Let

We only need to prove that (

In the proof of Theorem

Let

Since

By Theorem

Given a point

We devote this section to appling the obtained results in Section

Let

If

Suppose that

For the given

Let

If

Suppose that

Similar to the proof of Theorem

The authors are grateful to the two anonymous reviewers for their valuable comments and suggestions, which helped to improve the paper. This research was partially supported by the National Natural Science Foundation of China (Grant no. 11071267).