Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems

Abstract

By using the concept of Fréchet differentiability of mapping, we present the Kuhn-Tucker optimality conditions for weakly efficient solution, Henig efficient solution, superefficient solution, and globally efficient solution to the vector equilibrium problems with constraints.

1. Introduction

Recently, some authors have studied the optimality conditions for vector variational inequalities. Giannessi et al. [1] turned the vector variational inequalities with constraints into vector variational inequalities without constraints and gave sufficient conditions for the efficient solutions and the weakly efficient solutions to the vector variational inequalities in . By using the concept of subdifferential of the function, Morgan and Romaniello [2] gave the scalarization and Kuhn-Tucker-like conditions for the weak vector generalized quasivariational inequalities in Hilbert space. Yang and Zheng [3] gave the optimality conditions for approximate solutions of vector variational inequalities in Banach space. On the other hand, some authors have derived the optimality conditions for weakly efficient solutions to vector optimization problems (see [4–19]).

Vector variational inequality problems and vector optimization problems, as well as several other problems, are special realizations of vector equilibrium problems (see [20, 21]); therefore, it is important to give the optimality conditions for the solution to the vector equilibrium problems for in this way we can turn the vector equilibrium problem with constraints to a corresponding scalar optimization problem without constraints, and we can then determine if the solution of the scalar optimization problem is a solution of the original vector equilibrium problem. Under the assumption of convexity, Gong [22] investigated optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to vector equilibrium problems with constraints and obtained that the weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems with constraints are equivalent to solution of corresponding scalar optimization problems without constraints, respectively. Qiu [23] presented the necessary and sufficient conditions for globally efficient solution under generalized cone-subconvexlikeness. Gong and Xiong [24] weakened the convexity assumptions in [22] and obtained necessary and sufficient conditions for weakly efficient solution, too.

In this paper, by using the concept of Fréchet differentiability of mapping, we study the optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to the vector equilibrium problems. We give Kuhn-Tucker necessary conditions to the vector equilibrium problems without convexity conditions and Kuhn-Tucker sufficient conditions with convexity conditions.

2. Preliminaries and Definitions

Throughout this paper, let be a real normed space, let and be real partially ordered normed spaces, and let and be closed convex pointed cones with , where denotes the interior of the set .

Let and be the topological dual spaces of and , respectively. Let

(2.1)

be the dual cones of and , respectively. Denote the quasi-interior of by , that is

(2.2)

Let be a nonempty subset of . The cone hull of is defined as

(2.3)

Denote the closure of by and interior of by .

A nonempty convex subset of the convex cone is called a base of , if and . It is easy to see that if and only if has a base.

Let be a base of . Set

(2.4)

By the separation theorem of convex sets, we know .

Denote the closed unit ball of by . Suppose that has a base . Let . It is clear that . The will be used for the rest of the paper. For any , denote cone, then cl is a closed convex pointed cone, and , for all (see [25]).

Let be a nonempty open convex subset, and let , be mappings.

We define the constraint set

(2.5)

and consider the vector equilibrium problems with constraints (for short, VEPC): find such that

(2.6)

where is a convex cone in .

Definition 2.1.

If , a vector satisfying

(2.7)

is called a weakly efficient solution to the VEPC.

For each , we denote

(2.8)

Definition 2.2 (see [22]).

Let have a base . A vector is called a Henig efficient solution to the VEPC if there exists some such that

(2.9)

Definition 2.3 (see [22]).

A vector is called a globally efficient solution to the VEPC if there exists a point convex cone with such that

(2.10)

Definition 2.4 (see [22]).

A vector is called a superefficient solution to the VEPC if there exists such that

(2.11)

where is closed unit ball of .

Definition 2.5.

Let be a real linear space, and let be a real topological linear space. Let be a nonempty subset of , and let a mapping and some be given. If for some the limit

(2.12)

exists, then is called the Gâteaux derivative of at in the direction . If this limit exists for each direction , the mapping is called Gâteaux differentiable at .

Definition 2.6.

Let and be real normed spaces, and let be a nonempty open subset of . Moreover, let a mapping and some be given. If there exists a continuous linear mapping with the property

(2.13)

then is called the Fréchet derivative of at and is called Fréchet differentiable at .

Remark 2.7.

By [26, Lemma .18], we can see that if is Fréchet differentiable at , then is Gâteaux differentiable at and the Fréchet derivative of at is equal to the Gâteaux derivative of at in each direction .

Definition 2.8.

Let and be real linear spaces, let be a pointed convex cone in , and let be a nonempty convex subset of . A mapping is called -convex, if for all and all

(2.14)

Lemma 2.9 (see [18]).

Assume that pointed convex cone has a base ; then one has the following.

(i)For any , .

(ii)For any , there exists some with .

(iii), and when is bounded and closed, then , where is the interior of in with respect to the norm of .

3. Optimality Condition

In this section, we give the Kuhn-Tucker necessary conditions and Kuhn-Tucker sufficient conditions for weakly efficient solution, Henig efficient solution, globally efficient solution, and superefficient solution to the vector equilibrium problems with constraints.

Let be given. Denote the mapping by

(3.1)

By the proof of Theorem .20 in [26] or from the definition, we have the following lemma.

Lemma 3.1.

Let and be real normed spaces, let be a nonempty open convex subset of , and let be closed convex pointed cone in . Assume that is -convex and is Gâteaux differentiable at . Then

(3.2)

Theorem 3.2.

Let , , and be real normed spaces, and let and be closed convex pointed cones in and with and , respectively. Let . Assume that and are Fréchet differentiable at Furthermore, assume that there exists such that . If is a weakly efficient solution to the VEPC, then there exist , such that

(3.3)

Proof.

Assume that is a weakly efficient solution to the VEPC. Define the set

(3.4)

Since and are linear operators, we can see that is a nonempty open convex set. We claim that . If not, then there exists such that

(3.5)

From Remark 2.7, we obtain

(3.6)

Since and are open sets, there exists some such that

(3.7)

From , , and , we can see that

(3.8)

Since is a convex set, . Thus, we get

(3.9)

This contradicts that is a weakly efficient solution to the VEPC. Thus . Noting that is an open set, by the separation theorem of convex sets (see [26]), there exists such that

(3.10)

Let . Then there exists such that

(3.11)

It is clear that for every , , , we have and . By (3.10), we have

(3.12)

We can get

(3.13)

Since is a closed convex cone, . By the continuity of , we can see that for all . That is, . Similarly, we can show that . We also have . In fact, if , from (3.10) we get

(3.14)

By assumption, there exists such that ; thus, we have

(3.15)

Hence

(3.16)

In particular, we have , and we get . This is a contradiction. Thus, . It is clear that

(3.17)

for all , , By (3.10), we obtain

(3.18)

for all , ,

Letting , we get

(3.19)

It is clear that

(3.20)

for all , , . By (3.10), we have

(3.21)

Letting , we obtain . Noting that and , we have . Thus,

(3.22)

From (3.19) and (3.22), we have

(3.23)

Thus, we have

(3.24)

This completes the proof.

Remark 3.3.

The condition that there exists such that is a generalized Slater constraint condition. In fact, from the proof of Theorem 3.2, we obtain that there exists such that and .

On the other hand, if is Gâteaux differentiable at , and is -convex on , and then . In fact, by Lemma 3.1, there exists such that

(3.25)

Theorem 3.4.

Let , , and be real normed spaces, let be a closed convex pointed cone with , and let be a closed convex pointed cone with , and let . Assume that and are Gâteaux differentiable at , is -convex on , and is -convex on . If there exist , such that

(3.26)

(3.27)

then is a weakly efficient solution to the VEPC.

Proof.

Since the mappings and are Gâteaux differentiable at , is -convex on , and is -convex on , from Lemma 3.1, we have

(3.28)

From , and (3.26), we get

(3.29)

By (3.27), we have

(3.30)

We will show that is a weakly efficient solution to the VEPC. If not, then there exists such that

(3.31)

From and the above statement, we have

(3.32)

Noticing , we have ; so because of . Hence,

(3.33)

This contradicts (3.30). Hence, is a weakly efficient solution to the VEPC.

From Theorems 3.2 and 3.4, and Remark 2.7, we get the following corollary.

Corollary 3.5.

Let , , and be real normed spaces, let be a closed convex pointed cone with , and let be a closed convex pointed cone with , and let . Assume that and are Fréchet differentiable at , is -convex on , and is -convex on . Furthermore, assume that there exists such that . Then is a weakly efficient solution to the VEPC if and only if there exist , , such that

(3.34)

The concept of weakly efficient solution to the vector equilibrium problem requires the condition that the ordering cone has an nonempty interior. If the ordering cone has an empty interior, we cannot discuss the property of weakly efficient solutions to the vector equilibrium problem. However, if the ordering cone has a base, we can give necessary conditions and sufficient conditions for Henig efficient solutions and globally efficient solutions to the vector equilibrium problems with constraints.

Theorem 3.6.

Let , , and be real normed spaces, let be a closed convex pointed cone in with a base, let be a closed convex pointed cone in with , and let . Assume that and are Fréchet differentiable at . Furthermore, Assume that there exists such that . If is a Henig efficient solution to the VEPC, then there exist such that

(3.35)

Proof.

Assume that is a Henig efficient solution to the VEPC. By the definition, there exists some such that

(3.36)

Define the set

(3.37)

It is clear that is a nonempty open convex set and . By the separation theorem of convex sets, there exists such that

(3.38)

Let . Then there exists such that

(3.39)

Hence, for every , , , , we have and ; this implies that , . In a way similar to the proof of Theorem 3.2, we have . By Lemma 2.9, we can see that . It is clear that for all , , By (3.38), we get

(3.40)

In a way similar to the proof of Theorem 3.2, we have

(3.41)

From (3.40) and (3.41), we have

(3.42)

This completes the proof.

Theorem 3.7.

Let , , and be real normed spaces, let be a closed convex pointed cone in with a base, let be a closed convex pointed cone in with , and let . Assume that and are Gâteaux differentiable at , is -convex on , and is -convex on . If there exist such that

(3.43)

(3.44)

then is a Henig efficient solution to the VEPC.

Proof.

From , , (3.43), and (3.44), in a way similar to the proof of Theorem 3.4, we have

(3.45)

We will show that is a Henig efficient solution to the VEPC; that is, there exists some such that

(3.46)

Suppose to the contrary that for any ,

(3.47)

Then by , and by Lemma 2.9, there exists some such that . For this we have Thus, there exists such that

(3.48)

By , we have

(3.49)

Notice , we have , and thus we obtain because of . Hence,

(3.50)

This contradicts (3.45). Hence, is a Henig efficient solution to the VEPC.

Corollary 3.8.

Let , , and be real normed spaces, and be a closed convex pointed cone with a base, let be a closed convex pointed cone with , and let . Assume that and are Fréchet differentiable at , is -convex on , and is -convex on . Furthermore, assume that there exists such that . Then is a Henig efficient solution to the VEPC if and only if there exist , such that

(3.51)

Remark 3.9.

If has a bounded closed base , in view of Lemma 2.9, we have ; besides, is a superefficient solution to the VEPC if and only if is a Henig efficient solution to the VEPC (see [22]). Hence, by Corollary 3.8, we have the following corollary.

Corollary 3.10.

Let , , and be real normed spaces, let be a closed convex pointed cone with a bounded closed base, let be a closed convex pointed cone with , and let . Assume that and are Fréchet differentiable at , is -convex on , and is -convex on . Furthermore, assume that there exists such that . Then is a superefficient efficient solution to the VEPC if and only if there exists (with respect to the norm topology of ), such that

(3.52)

Theorem 3.11.

Let , , and be real normed spaces, let be a closed convex pointed cone in with a base, let be a closed convex pointed cone in with , and let . Assume that and are Fréchet differentiable at , and there exists such that . If is a globally efficient solution to the VEPC, then there exist , such that

(3.53)

Proof.

Assume that is a globally efficient solution to the VEPC. By the definition, there exists a pointed convex cone such that and

(3.54)

Define the set

(3.55)

Since is a convex cone, we know that is a nonempty open convex set and . By the separation theorem of convex sets, there exists such that

(3.56)

Let . Then there exists such that

(3.57)

Hence, for every , , , , we have and . This implies that , , and therefore because of . It is clear that

(3.58)

for all , , By (3.56), we get

(3.59)

for all , ,

Letting , we get

(3.60)

In a way similar to the proof of Theorem 3.2, we have

(3.61)

From (3.60) and (3.61), we have

(3.62)

This completes the proof.

Theorem 3.12.

Let , , and be real normed spaces, let be a closed convex pointed cone in with a base, let be a closed convex pointed cone in with , and let . Assume that and are Gâteaux differentiable at , is -convex on , and is -convex on . If there exist such that

(3.63)

(3.64)

then is a globally efficient solution to the VEPC.

Proof.

From , (3.63), and (3.64), in a way similar to the proof of Theorem 3.4, we have

(3.65)

We will show that is a globally efficient solution to the VEPC; that is, there exists a pointed convex cone such that and

(3.66)

Suppose to the contrary that for any pointed convex cone with , we have that

(3.67)

By , we set

(3.68)

We have , and is a pointed convex cone. By (3.67), there exists such that

(3.69)

By the definition of , we have that . Noticing , we have , and so because of . Hence

(3.70)

This contradicts (3.65). Hence, is a globally efficient solution to the VEPC.

Corollary 3.13.

Let , , and be real normed spaces, let be a closed convex pointed cone with a base, let be a closed convex pointed cone with , and let . Assume that and are Fréchet differentiable at , is -convex on , and is -convex on . Furthermore, assume that there exists such that . Then is a globally efficient solution to the VEPC if and only if there exist , such that

(3.71)