#### Abstract

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.

#### 1. Introduction

Let be Banach spaces, and let and be the origins of and , respectively. Let be a nonempty subset of and let be a closed convex pointed cone with apex at the origin. Let be partially ordered by and let be a bifunction. Consider the following vector equilibrium problem (for short, VEP): Recall from [1–4] that a vector is said to be a local efficient solution for VEP iff there exists a neighborhood of such that In particular, if , then is said to be an efficient solution for (VEP). In this paper, we pay our attention to the following stronger efficient solution, called local sharp efficient solution, for VEP.

*Definition 1. *A vector is said to be a local sharp efficient solution for (VEP) iff there exist a neighborhood of and a real number such that
where denotes the open unit ball of . particularly, if , then is said to be a sharp efficient solution for (VEP).

*Remark 2. *(i) Note that it always holds for several major classes of special problems, such as VOP, and VVI. Based on this fact, we consider all the points around in except in Definition 1 since .

(ii) If satisfying is a local sharp efficient solution for (VEP), then it is obvious that ; that is, is an isolated point of .

(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 [5–10] and references therein. Particularly, if we take for all with being a vector-valued map, then (VEP) reduces to the following vector optimization problem (for short, VOP): In the sequel, a vector is said to be a local sharp efficient solution (resp., sharp efficient solution) for VOP if and only if it is a local sharp efficient solution (resp., sharp efficient solution) for VEP; that is, there exist a neighborhood (resp., ) of and a real number such that where denotes the open ball with center at and radius . It is well known that the previous definition of local sharp efficient solution (resp., sharp efficient solution) for (VOP) is first proposed by Jiménez in [11], which is a natural generalization of the isolated local minima for scalar optimization problems. For more details, we refer to [12–17] and references therein. Analogously, if we take , where is a mapping and denotes the set of all linear continuous operators from to , then (VEP) reduces to the vector variational inequalities (for short, VVI) as follows: Moreover, a vector is said to be a local sharp efficient solution (resp., sharp efficient solution) for VVI if and only if it is local sharp efficient solution (resp., sharp efficient solution) for VEP; that is, there exist a neighborhood (resp., ) of and a real number such that

Recently, there has been increasing interest in dealing with optimality conditions for nonsmooth optimization problems by virtue of modern variational analysis techniques. Gong [4] established some necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems by using nonsmooth analysis. By means of convex analysis and nonsmooth analysis, Yang and Zheng [18] provided some sufficient conditions and necessary conditions for a point to be an approximate solution of vector variational inequalities. In [15], Zheng et al. studied sharp minima for multiobjective optimization problems in terms of the Mordukhovich coderivative and the normal cone and presented some optimality conditions. Moreover, Zhu et al. [19] extended the Fermat rules for the local minima of the constrained set-valued optimization problem to the sharp minima and the weak sharp minima in Banach spaces or Asplund spaces, by means of the Mordukhovich generalized differentiation and the normal cone.

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.

#### 2. Notations and Preliminaries

Throughout this paper, we denote by the interior of and the weak star topology on dual spaces. As usual, the distance function , the indicator function , and the support function for are, respectively, defined 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 [13, 20–23] and references therein for more details. Recall that the vector-valued map is said to be Fréchet differentiable at if there exists a linear continuous operator such that Here, is called the Fréchet derivative of at . As usual, we denote by the adjoint operator of ; that is, for all and . Moreover, is said to be strictly differentiable at if Let be a proper lower semicontinuous function. Recall that the Clarke-Rockafellar generalized directional derivative of at in the direction is defined by where means and . The Clarke subdifferential of at is defined as Furthermore, if is a convex function, then is in accordance with the subdifferential in convex analysis; that is, Given a point . Recall that the Clarke tangent cone of at is The Clarke normal cone of at is defined by the polar of ; that is, In particular, if is convex, then agrees with the normal cone of convex analysis; that is, and if is closed, then we have for all .

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

Proposition 3. *For every nonempty closed subset and every , one has and , where and denote the closed unit ball of and the -closure, respectively.*

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

Proposition 4 (generalized Fermat rule). *Let be a proper lower semicontinuous function. If attains a local minimum at , then .*

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

Proposition 5. *Let be proper lower semicontinuous functions and . If is locally Lipschitz around , then .*

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

Proposition 6. *Let be Banach spaces, Let be a vector-valued map, and Let be a real-valued function. Suppose that is strictly differentiable at and is locally Lipschitz around . Then is locally Lipschitz around , and one has
*

#### 3. Optimality Conditions

Given a point . In the sequel, let the vector-valued map be defined by

Theorem 7 (strong Fermat rule). *
Given a point with . Suppose that is a closed subset of . *(i)* If is a local sharp efficient solution for (VEP) and is strictly differentiable at , then one has
*(ii)* Let be convex and let be -convex on . Assume that is Fréchet differentiable at . Then it follows that
* *implies being a sharp efficient solution for (VEP). *

*Proof. *(i) Since is a local sharp efficient solution for (VEP), there exist a neighborhood of and a real number such that
which implies that
Take arbitrary . Then we have for all . Together with (22), it follows that is a local minimum point of the function defined by
By Proposition 4 (generalized Fermat rule), we have . Note that is strictly differentiable at and the distance function is globally Lipschitz with modulus 1. It follows from Proposition 6 that is locally Lipschitz around , and, moreover, together with Proposition 3, and , one has
Furthermore, is closed, which implies that is lower semicontinuous function. Thus, by Proposition 5 and (24), we have
Since is arbitrary, it follows that , which implies that

(ii) Since , there exists some real number such that
Take arbitrary . Then there exist and such that
Since is -convex on , we have ([24, Theorem 2.20])
Together with , we have
which implies that
Moreover, it follows from being convex and that for all . Together with (28) and (31), we can conclude that
Since is arbitrary, we have
that is,
Thus, is a sharp efficient solution for (VEP).

*Remark 8. *Assume that is convex and is Fréchet differentiable at . Then it is obvious that
Implies that
Since is a cone, we have
It is worth noting that (37) implies (35); that is, (35), (36), and (37) are equivalent. In the following lemma, we prove this assertion.

Lemma 9. *Let be convex and let be Fréchet differentiable at . Then (35), (36) and (37) are equivalent.*

*Proof. *We only need to prove that (37) implies (35). It follows from (37) that
Note that the dual space is complete. By (38) and the Baire's category theorem ([25, Theorem 4.7-2]) there exists some such that
Thus, we have . Moreover, we can conclude that
since is closed. In fact, take arbitrary sequence converging to some . Then, there exist and such that for all . Since the closed unit ball of the dual space of a Banach space is weak*-compact, without loss of generality, we can assume that
(passing to a subsequence if necessary). Moreover, since the conjugate operator is weak*-weak* continuous, we get
Since is convex, it follows from that
Thus, we have ; that is, . Together with (41) and (42), it follows that . Therefore, we have shown that is closed. By (40), there exist some and such that
Moreover, since , it follows from (38) that there exists some such that . Together with the closedness of , we have
Since and are convex cones, we get from (44) that
that is,
Thus, , which implies that
This completes the proof.

*Remark 10. *In the proof of Theorem 7(i), we have shown that if is strictly differentiable at and , then . Moreover, if, in addition, is -convex on , then we have
We give the following lemma to explain this result.

Lemma 11. *Let be strictly differentiable at and . Suppose that is -convex on ; then it follows that
*

*Proof. *Since is a convex cone, the real-valued function is monotonically increasing; that is, , implies that . Together with is -convex, we have that ([24, Lemma 2.7(b)]) being convex. By the proof of Theorem 7(i), it is sufficient to prove that . Since is a closed and convex cone, and , it follows that ([26, Theorem 3.1])
Note that is -convex on and Fréchet differentiable at , and . Then we get ([24, Theorem 2.20]) for all . Thus, we have
Then, for every , it follows that ; that is, . Since is arbitrary, we can conclude that .

By Theorem 7 and Lemmas 9 and 11, we immediately have the the following characterization of the sharp efficiency for (VEP) in convex case.

Corollary 12. *
Given a point with . Let be a closed and convex subset of , and let be -convex on . Suppose that is strictly differentiable at . Then the following assertions are equivalent: *(i)* is a local sharp efficient solution for (VEP),*(ii)* is a sharp efficient solution for (VEP),*(iii)*,*(iv)*,*(v)*,*(vi)*.*

#### 4. Applications

We devote this section to appling the obtained results in Section 3 to vector optimization problems and vector variational inequalities, respectively.

Theorem 13. *Let and let be a mapping. Suppose that is a closed subset of . *(i)* If is a local sharp efficient solution for (VOP) and is strictly differentiable at , then one has
*(ii)* Suppose that is convex, is strictly differentiable at and -convex on . Then the following assertions are equivalent:(a) is local sharp efficient solution for (VOP),(b) is sharp efficient solution for (VOP),(c),(d),(e).*

*Proof. *For the given , we take for all . Then is a local sharp efficient solution for (VOP) if and only if it is a local sharp efficient solution for (VEP). Moreover, is strictly differentiable at if and only if is strictly differentiable at . When is convex, the -convexity of on is equivalent to the -convexity of . Together with Theorem 7 and Corollary 12, we complete the proof.

Theorem 14. *Let and let be a mapping. Suppose that is a closed subset of . *(i)* If is a local sharp efficient solution for (VVI), then one has
*(ii)* Suppose that is convex. Then the following assertions are equivalent:(a) is local sharp efficient solution for (VVI),(b) is sharp efficient solution for (VVI),(c),(d),(e).*

*Proof. *Similar to the proof of Theorem 13, we take for the given and for all . Then is a local sharp efficient solution for (VVI) if and only if it is a local sharp efficient solution for (VEP). Moreover, since , is obviously strictly differentiable at and -convex on whenever is a convex subset of . Combined with Theorem 7 and Corollary 12, this completes the proof.

#### Acknowledgments

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).