Volume 2013, Issue 1 128178

Research Article

Open Access

Sharp Efficiency for Vector Equilibrium Problems on Banach Spaces

Si-Huan Li,

Corresponding Author

Si-Huan Li

[email protected]

School of Economics and Business Administration, Chongqing University, Chongqing 400030, China cqu.edu.cn

Department of Business Administration, Huaihua University, Huaihua, Hunan 418000, China hhtc.edu.cn

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Qiang Wang,

Qiang Wang

Automobile and Traffic Engineering College, Heilongjiang Institute of Technology, Harbin 150050, China

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Shu Xu,

Shu Xu

Rear Services Office, Chongqing Police College, Chongqing 401331, China

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Jun-Xiang Wang,

Jun-Xiang Wang

Office of Academic Affairs, Heilongjiang Institute of Technology, Harbin 150050, China

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Si-Huan Li,

Corresponding Author

Si-Huan Li

[email protected]

School of Economics and Business Administration, Chongqing University, Chongqing 400030, China cqu.edu.cn

Department of Business Administration, Huaihua University, Huaihua, Hunan 418000, China hhtc.edu.cn

Search for more papers by this author

Qiang Wang,

Qiang Wang

Automobile and Traffic Engineering College, Heilongjiang Institute of Technology, Harbin 150050, China

Search for more papers by this author

Shu Xu,

Shu Xu

Rear Services Office, Chongqing Police College, Chongqing 401331, China

Search for more papers by this author

Jun-Xiang Wang,

Jun-Xiang Wang

Office of Academic Affairs, Heilongjiang Institute of Technology, Harbin 150050, China

Search for more papers by this author

First published: 27 March 2013

https://doi.org/10.1155/2013/128178

Academic Editor: Qun Lin

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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):

()

Recall from [1–4] that a vector

is said to be a local efficient solution for VEP iff there exists a neighborhood U of

such that

()

In particular, if U = X, 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 U of and a real number η > 0 such that

()

where B_Y denotes the open unit ball of Y. particularly, if U = X, 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 S except in Definition 1 since 0_Y ∈ −C.

(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 f : X → Y 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 U (resp., U = X) of

and a real number η > 0 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 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:

()

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 U (resp., U = X) of

and a real number η > 0 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 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

()

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

if there exists a linear continuous operator

such that

()

Here,

is called the Fréchet derivative of F at

. As usual, we denote by

the adjoint operator of

; that is,

for all x ∈ X and y ∈ Y. Moreover, F is said to be strictly differentiable at

()

Let h : X → ℝ ∪ {+∞} be a proper lower semicontinuous function. Recall that the Clarke-Rockafellar generalized directional derivative

of h at

in the direction d ∈ X is defined by

()

where

means

and

. The Clarke subdifferential

of h at

is defined as

()

Furthermore, if h 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 S at

()

The Clarke normal cone

of S at

is defined by the polar of

; that is,

()

In particular, if S is convex, then

agrees with the normal cone

of convex analysis; that is,

()

and if S is closed, then we have

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 and N_c(Ω, x) = cl^* {⋃_λ>0 λ∂_cd(•, Ω)(x)}, where 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 , then .

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 ∈ dom f∩dom h. 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

()

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 S is a closed subset of X.

(i)
If is a local sharp efficient solution for (VEP) and is strictly differentiable at , then one has
()
(ii)
Let S be convex and let be C-convex on S. 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 U of and a real number η > 0 such that

()

which implies that

()

Take arbitrary

. Then we have

for all x ∈ X. Together with (22), it follows that

is a local minimum point of the function f : X → ℝ 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 N_c(−C, 0_Y) = C⁺, one has

()

Furthermore, S is closed, which implies that ψ 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 η > 0 such that

()

Take arbitrary

. Then there exist

and

such that

()

Since

is C-convex on S, we have ([24, Theorem 2.20])

()

Together with

, we have

()

which implies that

()

Moreover, it follows from S being convex and

that

for all x ∈ S. 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 S 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 S 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 X^* is complete. By (38) and the Baire′s category theorem ([25, Theorem 4.7-2]) there exists some n₀ ∈ ℕ such that

()

Thus, we have

. Moreover, we can conclude that

()

since

is closed. In fact, take arbitrary sequence

converging to some x^* ∈ X^*. Then, there exist

and

such that

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

()

(passing to a subsequence if necessary). Moreover, since the conjugate operator

is weak*-weak* continuous, we get

()

Since S 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 ρ > 0 such that

()

Moreover, since

, it follows from (38) that there exists some n₁ ∈ ℕ such that

. Together with the closedness of

, we have

()

Since C⁺ 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 C-convex on X, then we have

()

We give the following lemma to explain this result.

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

()

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 is C-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 C is a closed and convex cone, and (−C) ^∘ = C⁺, it follows that ([26, Theorem 3.1])

()

Note that

is C-convex on X and Fréchet differentiable at

, and

. Then we get ([24, Theorem 2.20])

for all x ∈ X. 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 S be a closed and convex subset of X, and let be C-convex on S. 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 f : X → Y be a mapping. Suppose that S is a closed subset of X.

(i)
If is a local sharp efficient solution for (VOP) and f is strictly differentiable at , then one has
()
(ii)
Suppose that S is convex, f is strictly differentiable at and C-convex on S. 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 x ∈ X. 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 f is strictly differentiable at . When S is convex, the C-convexity of on S is equivalent to the C-convexity of f. Together with Theorem 7 and Corollary 12, we complete the proof.

Theorem 14. Let and let T : X → 𝕃(X, Y) be a mapping. Suppose that S is a closed subset of X.

(i)
If is a local sharp efficient solution for (VVI), then one has
()
(ii)
Suppose that S 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 x ∈ X. 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 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).

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Sharp Efficiency for Vector Equilibrium Problems on Banach Spaces

Abstract

1. Introduction

2. Notations and Preliminaries

3. Optimality Conditions

4. Applications

Acknowledgments

References

References

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Sharp Efficiency for Vector Equilibrium Problems on Banach Spaces

Abstract

1. Introduction

2. Notations and Preliminaries

3. Optimality Conditions

4. Applications

Acknowledgments

References

References

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