Sharp Efficiency for Vector Equilibrium Problems on Banach Spaces

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 𝑋,𝑌 be Banach spaces, and let 0_𝑋 and 0_𝑌 be the origins of 𝑋 and 𝑌, respectively. Let 𝑆 be a nonempty subset of 𝑋 and 𝐶⊂𝑌 let be a closed convex pointed cone with apex at the origin. Let 𝑌 be partially ordered by 𝐶 and let 𝐹∶𝑋×𝑋→𝑌 be a bifunction. Consider the following vector equilibrium problem (for short, VEP): find𝑥∈𝑆suchthat𝐹(𝑥,𝑥)∉−𝐶⧵0_𝑌∀𝑥∈𝑆. (1) Recall from [1–4] that a vector 𝑥∈𝑆 is said to be a local efficient solution for VEP iff there exists a neighborhood 𝑈 of 𝑥 such that 𝐹(𝑥,𝑥)∉−𝐶⧵0_𝑌,∀𝑥∈𝑆∩𝑈, (2) In particular, if 𝑈=𝑋, then 𝑥 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 𝑥∈𝑆 is said to be a local sharp efficient solution for (VEP) iff there exist a neighborhood 𝑈 of 𝑥 and a real number 𝜂>0 such that 𝐹(𝑥,𝑥)∉−𝐶+𝜂‖𝑥−𝑥‖𝐵_𝑌,∀𝑥∈𝑆∩𝑈⧵{𝑥}, (3) where 𝐵_𝑌 denotes the open unit ball of 𝑌. particularly, if 𝑈=𝑋, then 𝑥 is said to be a sharp efficient solution for (VEP).

Remark 2. (i) Note that it always holds 𝐹(𝑥,𝑥)=0_𝑌 for several major classes of special problems, such as VOP, and VVI. Based on this fact, we consider all the points around 𝑥 in 𝑆 except 𝑥 in Definition 1 since 0_𝑌∈−𝐶.
(ii) If 𝑥∈𝑆 satisfying 𝐹(𝑥,𝑥)=0_𝑌 is a local sharp efficient solution for (VEP), then it is obvious that 𝑆∩𝑈={𝑥}; that is, 𝑥 is an isolated point of 𝑆.
(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 𝐹(𝑥,𝑥)=𝑓(𝑥)−𝑓(𝑥) for all 𝑥,𝑥∈𝑋 with 𝑓∶𝑋→𝑌 being a vector-valued map, then (VEP) reduces to the following vector optimization problem (for short, VOP): min𝑓(𝑥),s.t.𝑥∈𝑆. (4) In the sequel, a vector 𝑥∈𝑆 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 𝑈 (resp., 𝑈=𝑋) of 𝑥 and a real number 𝜂>0 such that 𝑓(𝑥)𝑓(𝑥)‖𝑥‖{𝑥}+𝐶∩ℬ,𝜂𝑥−=Ø,∀𝑥∈𝑆∩𝑈⧵ (5) where ℬ(𝑓(𝑥),𝜂‖𝑥−𝑥‖) denotes the open ball with center at 𝑓(𝑥) and radius 𝜂‖𝑥−𝑥‖. 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 𝐹(𝑥,𝑥)=⟨𝑇(𝑥),𝑥−𝑥⟩, where 𝑇∶𝑋→𝕃(𝑋,𝑌) is a mapping and 𝕃(𝑋,𝑌) denotes the set of all linear continuous operators from 𝑋 to 𝑌, then (VEP) reduces to the vector variational inequalities (for short, VVI) as follows: find𝑥∈𝑆suchthat⟨𝑇(𝑥),𝑥−𝑥⟩∉−𝐶⧵0_𝑌∀𝑥∈𝑆. (6) Moreover, a vector 𝑥∈𝑆 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 𝑈 (resp., 𝑈=𝑋) of 𝑥 and a real number 𝜂>0 such that ⟨𝑇(𝑥),𝑥−𝑥⟩∉−𝐶+𝜂‖𝑥−𝑥‖𝐵_𝑌,∀𝑥∈𝑆∩𝑈⧵{𝑥}. (7)

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 int𝑆 the interior of 𝑆 and 𝑤^∗ the weak star topology on dual spaces. As usual, the distance function 𝑑(•,𝑆)∶𝑋→ℝ, the indicator function 𝜓(•,𝑆)∶𝑋→ℝ∪{+∞}, and the support function 𝜓^∗(•,𝑆)∶𝑋^∗→ℝ∪{+∞} for 𝑆 are, respectively, defined by 𝑑(𝑥,𝑆)∶=inf_𝑦∈𝑆𝜓()()𝜓𝑥−𝑦,∀𝑥∈𝑋,𝑥,𝑆∶=0if𝑥∈𝑆,𝜓𝑥,𝑆∶=+∞,if𝑥∉𝑆,_∗𝑥^∗,𝑆∶=sup_𝑥∈𝑆⟨𝑥^∗,𝑥⟩,∀𝑥^∗∈𝑋^∗. (8)

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 𝐹∶𝑋→𝑌 is said to be Fréchet differentiable at 𝑥 if there exists a linear continuous operator ∇𝐹(𝑥)∶𝑋→𝑌 such that lim_𝑥→𝑥𝐹(𝑥)−𝐹(𝑥)−∇𝐹(𝑥)(𝑥−𝑥)‖𝑥−𝑥‖=0_𝑌. (9) Here, ∇𝐹(𝑥) is called the Fréchet derivative of 𝐹 at 𝑥. As usual, we denote by (∇𝐹(𝑥))^∗∶𝑌^∗→𝑋^∗ the adjoint operator of ∇𝐹(𝑥); that is, ⟨∇𝐹(𝑥)(𝑥),𝑦⟩=⟨𝑥,(∇𝐹(𝑥))^∗(𝑦)⟩ for all 𝑥∈𝑋 and 𝑦∈𝑌. Moreover, 𝐹 is said to be strictly differentiable at 𝑥 if lim_𝑥→𝑥^,𝑢→𝑥^,𝑥≠𝑢𝐹(𝑥)−𝐹(𝑢)−∇𝐹(𝑥)(𝑥−𝑢)‖𝑥−𝑢‖=0_𝑌. (10) Let ℎ∶𝑋→ℝ∪{+∞} be a proper lower semicontinuous function. Recall that the Clarke-Rockafellar generalized directional derivative ℎ^↑(𝑥,𝑑) of ℎ at 𝑥∈domℎ∶={𝑥∈𝑋∣ℎ(𝑥)<+∞} in the direction 𝑑∈𝑋 is defined by ℎ^↑(𝑥,𝑑)∶=lim_𝜖↓0limsup_𝑥ℎ→−𝑥_,𝑡↓0inf_{𝑢∈𝑑+𝜖𝐵𝑋}ℎ(𝑥+𝑡𝑢)−𝑓(𝑥)𝑡, (11) where 𝑥→−^ℎ𝑥 means 𝑥→𝑥 and ℎ(𝑥)→ℎ(𝑥). The Clarke subdifferential 𝜕_𝑐ℎ(𝑥) of ℎ at 𝑥∈domℎ is defined as 𝜕_𝑐ℎ(𝑥)∶=𝑥^∗∈𝑋^∗∣𝑥^∗,𝑑≤ℎ^↑(𝑥,𝑑),∀𝑑∈𝑋. (12) Furthermore, if ℎ is a convex function, then 𝜕_𝑐ℎ(𝑥) is in accordance with the subdifferential 𝜕ℎ(𝑥) in convex analysis; that is, 𝜕_𝑐ℎ(𝑥)(𝑥)=𝑥=𝜕ℎ_∗∈ℝ^𝑛∣𝑥^∗,𝑥−𝑥≤ℎ(𝑥)−ℎ(𝑥),∀𝑥∈𝑋. (13)Given a point 𝑥∈𝑆. Recall that the Clarke tangent cone 𝑇_𝑐(𝑆,𝑥) of 𝑆 at 𝑥 is 𝑇_𝑐(𝑆,𝑥)∶=𝑣∈𝑋∣∀𝑥_𝑛⟶𝑥,∀𝑡_𝑛⟶0⁺,∃𝑣_𝑛⟶𝑣s.t.𝑥_𝑛+𝑡_𝑛𝑣_𝑛∈𝑆,∀𝑛∈ℕ. (14) The Clarke normal cone 𝑁_𝑐(𝑆,𝑥) of 𝑆 at 𝑥 is defined by the polar of 𝑇_𝑐(𝑆,𝑥); that is, 𝑁_𝑐(𝑆,𝑥)∶=𝑇_𝑐(𝑆,𝑥)^∘∶=𝑥^∗∈𝑋^∗∣⟨𝑥^∗,𝑣⟩≤0,∀𝑣∈𝑇_𝑐(𝑆,𝑥). (15) In particular, if 𝑆 is convex, then 𝑁_𝑐(𝑆,𝑥) agrees with the normal cone 𝑁(𝑆,𝑥) of convex analysis; that is, 𝑁_𝑐(𝑥)(𝑥)=𝑥𝑆,=𝑁𝑆,_∗∈𝑋^∗∣⟨𝑥^∗,𝑥−𝑥⟩≤0,∀𝑥∈𝑆, (16) and if 𝑆 is closed, then we have 𝜕_𝑐𝜓(𝑥,𝑆)=𝑁_𝑐(𝑆,𝑥) for all 𝑥∈𝑆.

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

Proposition 3. For every nonempty closed subset Ω⊂𝑋 and every 𝑥∈Ω, one has 𝜕_𝑐𝑑(•,Ω)(𝑥)⊂𝔹_𝑋^∗∩𝑁_𝑐(Ω,𝑥) and 𝑁_𝑐(Ω,𝑥)=cl^∗{⋃_𝜆>0𝜆𝜕_𝑐𝑑(•,Ω)(𝑥)}, where 𝔹_𝑋^∗ and cl^∗ denote the closed unit ball of 𝑋^∗ and the 𝑤^∗-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 𝑓∶𝑋→ℝ∪{+∞} be a proper lower semicontinuous function. If 𝑓 attains a local minimum at 𝑥∈𝑋, then 0_𝑋^∗∈𝜕_𝑐𝑓(𝑥).

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

Proposition 5. Let 𝑓,ℎ∶𝑋→ℝ∪{+∞} be proper lower semicontinuous functions and 𝑥∈dom𝑓∩domℎ. If 𝑓 is locally Lipschitz around 𝑥, then 𝜕_𝑐(𝑓+ℎ)(𝑥)⊂𝜕_𝑐𝑓(𝑥)+𝜕_𝑐ℎ(𝑥).

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

Proposition 6. Let 𝑋,𝑌 be Banach spaces, Let 𝐺∶𝑋→𝑌 be a vector-valued map, and Let 𝑔∶𝑌→ℝ∪{+∞} be a real-valued function. Suppose that 𝐺 is strictly differentiable at 𝑥 and 𝑔 is locally Lipschitz around 𝐺(𝑥). Then 𝑓=𝑔∘𝐺∶𝑋→ℝ∪{+∞} is locally Lipschitz around 𝑥, and one has 𝜕_𝑐𝑓(𝑥)⊂𝐺^′(𝑥)^∗𝜕_𝑐𝑔(𝐺(𝑥)). (17)

3. Optimality Conditions

Given a point 𝑥∈𝑆. In the sequel, let the vector-valued map 𝐹𝑥∶𝑋→𝑌 be defined by 𝐹𝑥(𝑥)∶=𝐹(𝑥,𝑥),∀𝑥∈𝑋. (18)

Theorem 7 (strong Fermat rule). Given a point 𝑥∈𝑆 with 𝐹(𝑥,𝑥)=0_𝑌. Suppose that 𝑆 is a closed subset of 𝑋. (i) If 𝑥 is a local sharp efficient solution for (VEP) and 𝐹𝑥 is strictly differentiable at 𝑥, then one has 0_𝑋^∗∈int∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗+𝑁_𝑐(𝑆,𝑥). (19)(ii) Let 𝑆 be convex and let 𝐹𝑥 be 𝐶-convex on 𝑆. Assume that 𝐹𝑥 is Fréchet differentiable at 𝑥. Then it follows that 0_𝑋^∗∈int∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗+𝑁(𝑆,𝑥) (20) implies 𝑥 being a sharp efficient solution for (VEP).

Proof. (i) Since 𝑥∈𝑆 is a local sharp efficient solution for (VEP), there exist a neighborhood 𝑈 of 𝑥 and a real number 𝜂>0 such that 𝐹(𝑥,𝑥)∉−𝐶+𝜂‖𝑥−𝑥‖𝐵_𝑌,∀𝑥∈𝑆∩𝑈⧵{𝑥}, (21) which implies that 𝑑𝐹𝑥(𝑥),−𝐶+𝜓(𝑥,𝑆)≥𝜂‖𝑥−𝑥‖,∀𝑥∈𝑈. (22) Take arbitrary 𝑥^∗∈𝔹_𝑋^∗. Then we have ⟨𝑥^∗,𝑥−𝑥⟩≤‖𝑥−𝑥‖ for all 𝑥∈𝑋. Together with (22), it follows that 𝑥∈𝑆 is a local minimum point of the function 𝑓∶𝑋→ℝ defined by 𝑓(𝑥)∶=−𝜂𝑥^∗,𝑥−𝑥+𝑑𝐹𝑥(𝑥),−𝐶+𝜓(𝑥,𝑆),∀𝑥∈𝑋. (23) By Proposition 4 (generalized Fermat rule), we have 0_𝑋^∗∈𝜕𝑓(𝑥). Note that 𝐹𝑥 is strictly differentiable at 𝑥 and the distance function is globally Lipschitz with modulus 1. It follows from Proposition 6 that 𝑑(𝐹𝑥(•),−𝐶)∶𝑋→ℝ is locally Lipschitz around 𝑥, and, moreover, together with Proposition 3, 𝐹(𝑥,𝑥)=0_𝑌 and 𝑁_𝑐(−𝐶,0_𝑌)=𝐶⁺, one has 𝜕_𝑐𝑑𝐹𝑥(•),−𝐶(𝑥)⊂∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗. (24) Furthermore, 𝑆 is closed, which implies that 𝜓 is lower semicontinuous function. Thus, by Proposition 5 and (24), we have 𝜂𝑥^∗∈∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗+𝑁_𝑐(𝑆,𝑥). (25) Since 𝑥^∗∈𝔹_𝑋^∗ is arbitrary, it follows that 𝜂𝔹_𝑋^∗⊂(∇𝐹𝑥(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗)+𝑁_𝑐(𝑆,𝑥), which implies that 0_𝑋^∗∈int∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗+𝑁_𝑐(𝑆,𝑥). (26)
(ii) Since 0_𝑋^∗∈int((∇𝐹𝑥(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗)+𝑁(𝑆,𝑥)), there exists some real number 𝜂>0 such that 𝜂𝔹_𝑋^∗⊂∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗+𝑁(𝑆,𝑥). (27) Take arbitrary 𝑥^∗∈𝔹_𝑋^∗. Then there exist 𝑦^∗∈𝐶⁺∩𝔹_𝑌^∗ and 𝑧^∗∈𝑁(𝑆,𝑥) such that 𝜂𝑥^∗=∇𝐹𝑥(𝑥)^∗𝑦^∗+𝑧^∗. (28) Since 𝐹𝑥 is 𝐶-convex on 𝑆, we have ([24, Theorem 2.20]) 𝐹𝑥(𝑥)+∇𝐹𝑥(𝑥)(𝑥−𝑥)∈𝐹𝑥(𝑥)−𝐶,∀𝑥∈𝑆. (29) Together with 𝑦^∗∈𝐶⁺∩𝔹_𝑌^∗, we have ∇𝐹𝑥(𝑥)^∗𝑦^∗𝑥=𝑦,𝑥−_∗𝑥(𝑥𝑥≤𝑦,∇𝐹)(𝑥−)_∗𝑥(𝑥≤𝑦,𝐹)_∗𝑥(𝑥)≤𝐹𝑥(𝑥),𝐹+𝑐+𝑐,∀𝑐∈𝐶,∀𝑥∈𝑆, (30) which implies that ∇𝐹𝑥(𝑥)^∗𝑦^∗,𝑥−𝑥≤𝑑𝐹𝑥(𝑥),−𝐶,∀𝑥∈𝑆. (31) Moreover, it follows from 𝑆 being convex and 𝑧^∗∈𝑁(𝑆,𝑥) that ⟨𝑧^∗,𝑥−𝑥⟩≤0 for all 𝑥∈𝑆. Together with (28) and (31), we can conclude that 𝜂𝑥^∗,𝑥−𝑥≤𝑑𝐹𝑥(𝑥),−𝐶,∀𝑥∈𝑆. (32) Since 𝑥^∗∈𝔹_𝑋^∗ is arbitrary, we have 𝜂‖𝑥−𝑥‖=sup_{𝑥∗∈𝔹𝑋∗}⟨𝜂𝑥^∗𝐹𝑥(𝑥),𝑥−𝑥⟩≤𝑑,−𝐶,∀𝑥∈𝑆, (33) that is, 𝐹(𝑥,𝑥)∉−𝐶+𝜂‖𝑥−𝑥‖𝐵_𝑌,∀𝑥∈𝑆⧵{𝑥}. (34) Thus, 𝑥 is a sharp efficient solution for (VEP).

Remark 8. Assume that 𝑆 is convex and 𝐹𝑥 is Fréchet differentiable at 𝑥. Then it is obvious that 0_𝑋^∗∈int∇𝐹𝑥(𝑥)^∗C⁺∩𝔹_𝑌^∗+𝑁(𝑆,𝑥) (35) Implies that 0_𝑋^∗∈int∇𝐹𝑥(𝑥)^∗𝐶⁺+𝑁(𝑆,𝑥). (36) Since (∇𝐹𝑥(𝑥))^∗(𝐶⁺)+𝑁(𝑆,𝑥) is a cone, we have ∇𝐹𝑥(𝑥)^∗𝐶⁺+𝑁(𝑆,𝑥)=𝑋^∗. (37) 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 𝑆 be convex and let 𝐹𝑥 be Fréchet differentiable at 𝑥. Then (35), (36) and (37) are equivalent.

Proof. We only need to prove that (37) implies (35). It follows from (37) that 𝑋^∗=^+∞_𝑛=1cl∇𝐹𝑥(𝑥)^∗𝑛𝔹_𝑌^∗∩𝐶⁺+𝑛𝔹_𝑋^∗∩𝑁(𝑆,𝑥). (38) Note that the dual space 𝑋^∗ is complete. By (38) and the Baire's category theorem ([25, Theorem 4.7-2]) there exists some 𝑛₀∈ℕ such that intcl∇𝐹𝑥(𝑥)^∗𝑛₀𝔹_𝑌^∗∩𝐶⁺+𝑛₀𝔹_𝑋^∗∩𝑁(𝑆,𝑥)≠Ø. (39) Thus, we have intcl((∇𝐹𝑥(𝑥))^∗(𝔹_𝑌^∗∩𝐶⁺)+𝔹_𝑋^∗∩𝑁(𝑆,𝑥))≠Ø. Moreover, we can conclude that int∇𝐹𝑥(𝑥)^∗𝔹_𝑌^∗∩𝐶⁺+𝔹_𝑋^∗∩𝑁(𝑆,𝑥)≠Ø (40) since (∇𝐹𝑥(𝑥))^∗(𝔹_𝑌^∗∩𝐶⁺)+𝔹_𝑋^∗∩𝑁(𝑆,𝑥) is closed. In fact, take arbitrary sequence {𝑥^∗_𝑛}⊂(∇𝐹𝑥(𝑥))^∗(𝔹_𝑌^∗∩𝐶⁺)+𝔹_𝑋^∗∩𝑁(𝑆,𝑥) converging to some 𝑥^∗∈𝑋^∗. Then, there exist 𝑢^∗_𝑛∈𝔹_𝑌^∗∩𝐶⁺ and 𝑣^∗_𝑛∈𝔹_𝑋^∗∩𝑁(𝑆,𝑥) such that 𝑥^∗_𝑛=(∇𝐹𝑥(𝑥))^∗(𝑢^∗_𝑛)+𝑣^∗_𝑛 for all 𝑛∈ℕ. Since the closed unit ball of the dual space of a Banach space is weak*-compact, without loss of generality, we can assume that 𝑢−−→^∗_𝑛^𝑤∗𝑢^∗∈𝔹_𝑌^∗∩𝐶⁺,𝑣−−→^∗_𝑛^𝑤∗𝑣^∗∈𝔹_𝑋^∗ (41) (passing to a subsequence if necessary). Moreover, since the conjugate operator (∇𝐹𝑥(𝑥))^∗ is weak*-weak* continuous, we get 𝑥^∗=∇𝐹𝑥(𝑥)^∗𝑢^∗+𝑣^∗. (42) Since 𝑆 is convex, it follows from 𝑣^∗_𝑛∈𝔹_𝑋^∗∩𝑁(𝑆,𝑥) that 𝑣^∗_𝑛,𝑥−𝑥≤0,∀𝑛∈ℕ,∀𝑥∈𝑆. (43) Thus, we have ⟨𝑣^∗,𝑥−𝑥⟩≤0,∀𝑥∈𝑆; that is, 𝑣^∗∈𝑁(𝑆,𝑥). Together with (41) and (42), it follows that 𝑥^∗∈(∇𝐹𝑥(𝑥))^∗(𝔹_𝑌^∗∩𝐶⁺)+𝔹_𝑋^∗∩𝑁(𝑆,𝑥). Therefore, we have shown that (∇𝐹𝑥(𝑥))^∗(𝔹_𝑌^∗∩𝐶⁺)+𝔹_𝑋^∗∩𝑁(𝑆,𝑥) is closed. By (40), there exist some 𝑥^∗₀∈𝑋^∗ and 𝜌>0 such that 𝑥^∗₀+𝜌𝔹_𝑋^∗⊂∇𝐹𝑥(𝑥)^∗𝔹_𝑌^∗∩𝐶⁺+𝔹_𝑋^∗∩𝑁(𝑆,𝑥). (44) Moreover, since −𝑥^∗₀∈𝑋^∗, it follows from (38) that there exists some 𝑛₁∈ℕ such that −𝑥^∗₀∈cl((∇𝐹x(𝑥))^∗(𝑛₁𝔹_𝑌^∗∩𝐶⁺)+𝑛₁𝔹_𝑋^∗∩𝑁(𝑆,𝑥)). Together with the closedness of (∇𝐹𝑥(𝑥))^∗(𝔹_𝑌^∗∩𝐶⁺)+𝔹_𝑋^∗∩𝑁(𝑆,𝑥), we have −1𝑛₁𝑥^∗₀∈∇𝐹𝑥(𝑥)^∗𝔹_𝑌^∗∩𝐶⁺+𝔹_𝑋^∗∩𝑁(𝑆,𝑥). (45) Since 𝐶⁺ and 𝑁(𝑆,𝑥) are convex cones, we get from (44) that 𝜌𝑛₁𝔹_𝑋^∗⊂11+𝑛₁×∇𝐹𝑥(𝑥)^∗𝔹_𝑌^∗∩𝐶⁺+𝔹_𝑋^∗∩𝑁(𝑆,𝑥), (46) that is, 𝜌𝑛₁𝔹+1^𝑋∗⊂∇𝐹𝑥(𝑥)^∗𝔹_𝑌^∗∩𝐶⁺+𝔹_𝑋^∗∩𝑁(𝑆,𝑥). (47) Thus, 0_𝑋^∗∈int((∇𝐹𝑥(𝑥))^∗(𝔹_𝑌^∗∩𝐶⁺)+𝔹_𝑋^∗∩𝑁(𝑆,𝑥)), which implies that 0_𝑋^∗∈int∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗+𝑁(𝑆,𝑥). (48) This completes the proof.

Remark 10. In the proof of Theorem 7(i), we have shown that if 𝐹𝑥 is strictly differentiable at 𝑥 and 𝐹(𝑥,𝑥)=0_𝑌, then 𝜕_𝑐𝑑(𝐹𝑥(•),−𝐶)(𝑥)⊂(∇𝐹𝑥(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗). Moreover, if, in addition, 𝐹𝑥 is 𝐶-convex on 𝑋, then we have 𝜕_𝑐𝑑𝐹𝑥(•),−𝐶(𝑥)=∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗. (49) We give the following lemma to explain this result.

Lemma 11. Let 𝐹𝑥 be strictly differentiable at 𝑥 and 𝐹(𝑥,𝑥)=0_𝑌. Suppose that 𝐹𝑥 is 𝐶-convex on 𝑋; then it follows that 𝜕𝑑𝐹𝑥(•),−𝐶(𝑥)=∇𝐹𝑥(𝑥)^∗𝐶⁺∩𝔹_𝑌^∗. (50)

Proof. Since 𝐶 is a convex cone, the real-valued function 𝑑(•,−𝐶)∶𝑌→ℝ is monotonically increasing; that is, ∀𝑦₁,𝑦₂∈𝑌, 𝑦₁≤_𝐶𝑦₂ implies that 𝑑(𝑦₁,−𝐶)≤𝑑(𝑦₂,−𝐶). Together with 𝐹𝑥 is 𝐶-convex, we have that ([24, Lemma 2.7(b)]) 𝑑(𝐹𝑥(•),−𝐶)∶𝑋→ℝ being convex. By the proof of Theorem 7(i), it is sufficient to prove that (∇𝐹𝑥(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗)⊂𝜕𝑑(𝐹𝑥(•),−𝐶)(𝑥). Since 𝐶 is a closed and convex cone, and (−𝐶)^∘=𝐶⁺, it follows that ([26, Theorem 3.1]) 𝑑𝐹𝑥(𝑥),−𝐶=𝜓^∗𝐹𝑥(𝑥),𝐶⁺∩𝔹_𝑌^∗=sup_{𝑢∗∈𝐶+∩𝔹𝑌∗}𝑢^∗,𝐹𝑥(𝑥),∀𝑥∈𝑋. (51) Note that 𝐹𝑥 is 𝐶-convex on 𝑋 and Fréchet differentiable at 𝑥, and 𝐹(𝑥,𝑥)=0_𝑌. Then we get ([24, Theorem 2.20]) ∇𝐹𝑥(𝑥−𝑥)∈𝐹𝑥(𝑥)−𝐶 for all 𝑥∈𝑋. Thus, we have sup_{𝑢∗∈𝐶+∩𝔹𝑌∗}𝑢^∗,∇𝐹𝑥(𝑥)(𝑥−𝑥)≤sup_{𝑢∗∈𝐶+∩𝔹𝑌∗}𝑢^∗𝑥(𝑥𝐹𝑥(𝑥)𝐹𝑥(𝑥),𝐹)=𝑑,−𝐶−𝑑,−𝐶,∀𝑥∈𝑋. (52) Then, for every 𝑢^∗∈𝐶⁺∩𝔹_𝑌^∗, it follows that ⟨𝑢^∗,∇𝐹𝑥(𝑥)(𝑥−𝑥)⟩=⟨(∇𝐹𝑥(𝑥))^∗𝑢^∗,𝑥−𝑥⟩≤𝑑(𝐹𝑥(𝑥),−𝐶)−𝑑(𝐹𝑥(𝑥),−𝐶),∀𝑥∈𝑋; that is, 𝑢^∗∈𝜕𝑑(𝐹𝑥(•),−𝐶)(𝑥). Since 𝑢^∗∈𝐶⁺∩𝔹_𝑌^∗ is arbitrary, we can conclude that (∇𝐹𝑥(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗)⊂𝜕𝑑(𝐹𝑥(•),−𝐶)(𝑥).

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 𝑥∈𝑆 with 𝐹(𝑥,𝑥)=0_𝑌. Let 𝑆 be a closed and convex subset of 𝑋, and let 𝐹𝑥 be 𝐶-convex on 𝑆. Suppose that 𝐹𝑥 is strictly differentiable at 𝑥. Then the following assertions are equivalent: (i)𝑥 is a local sharp efficient solution for (VEP),(ii)𝑥 is a sharp efficient solution for (VEP),(iii)0_𝑋^∗∈int(𝜕𝑑(𝐹𝑥(•),−𝐶)(𝑥)+𝑁(𝑆,𝑥)),(iv)0_𝑋^∗∈int((∇𝐹𝑥(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗)+𝑁(𝑆,𝑥)),(v)0_𝑋^∗∈int((∇𝐹𝑥(𝑥))^∗(𝐶⁺)+𝑁(𝑆,𝑥)),(vi)(∇𝐹𝑥(𝑥))^∗(𝐶⁺)+𝑁(𝑆,𝑥)=𝑋^∗.

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 𝑥∈𝑆 and let 𝑓∶𝑋→𝑌 be a mapping. Suppose that 𝑆 is a closed subset of 𝑋. (i) If 𝑥 is a local sharp efficient solution for (VOP) and 𝑓 is strictly differentiable at 𝑥, then one has 0_X^∗∈int(∇𝐹(𝑥))^∗𝐶⁺∩𝔹_𝑌^∗+𝑁_𝑐(𝑆,𝑥). (53)(ii) Suppose that 𝑆 is convex, 𝑓 is strictly differentiable at 𝑥 and 𝐶-convex on 𝑆. Then the following assertions are equivalent:(a)𝑥 is local sharp efficient solution for (VOP),(b)𝑥 is sharp efficient solution for (VOP),(c)0_𝑋^∗∈int((∇𝐹(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗)+𝑁(𝑆,𝑥)),(d)0_𝑋^∗∈int((∇𝐹(𝑥))^∗(𝐶⁺)+𝑁(𝑆,𝑥)),(e)(∇𝐹(𝑥))^∗(𝐶⁺)+𝑁(𝑆,𝑥)=𝑋^∗.

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

Theorem 14. Let 𝑥∈𝑆 and let 𝑇∶𝑋→𝕃(𝑋,𝑌) be a mapping. Suppose that 𝑆 is a closed subset of 𝑋. (i) If 𝑥 is a local sharp efficient solution for (VVI), then one has 0_𝑋^∗∈int(𝑇(𝑥))^∗𝐶⁺∩𝔹_𝑌^∗+𝑁_𝑐(𝑆,𝑥). (54)(ii) Suppose that 𝑆 is convex. Then the following assertions are equivalent:(a)𝑥 is local sharp efficient solution for (VVI),(b)𝑥 is sharp efficient solution for (VVI),(c)0_𝑋^∗∈int((𝑇(𝑥))^∗(𝐶⁺∩𝔹_𝑌^∗)+𝑁(𝑆,𝑥)),(d)0_𝑋^∗∈int((𝑇(𝑥))^∗(𝐶⁺)+𝑁(𝑆,𝑥)),(e)(𝑇(𝑥))^∗(𝐶⁺)+𝑁(𝑆,𝑥)=𝑋^∗.

Proof. Similar to the proof of Theorem 13, we take 𝐹(𝑥,𝑥)=⟨𝑇(𝑥),𝑥−𝑥⟩ for the given 𝑥∈𝑆 and for all 𝑥∈𝑋. Then 𝑥 is a local sharp efficient solution for (VVI) if and only if it is a local sharp efficient solution for (VEP). Moreover, since 𝑇(𝑥)∈𝕃(𝑋,𝑌), 𝐹𝑥 is obviously strictly differentiable at 𝑥 and 𝐶-convex on 𝑆 whenever 𝑆 is a convex subset of 𝑋. 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).