# Optimality Conditions of Globally Efficient Solution for Vector Equilibrium Problems with Generalized Convexity

- 435 Downloads
- 4 Citations

## Abstract

We study optimality conditions of globally efficient solution for vector equilibrium problems with generalized convexity. The necessary and sufficient conditions of globally efficient solution for the vector equilibrium problems are obtained. The Kuhn-Tucker condition of globally efficient solution for vector equilibrium problems is derived. Meanwhile, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems with constraints.

### Keywords

Convex Cone Efficient Solution Topological Vector Space Vector Optimization Problem Vector Variational Inequality## 1. Introduction

which is assumed to be nonempty.

where Open image in new window is a convex cone in Open image in new window .

Vector equilibrium problems, which contain vector optimization problems, vector variational inequality problems, and vector complementarity problems as special case, have been studied by Ansari et al. [2, 3], Bianchi et al. [4], Fu [5], Gong [6], Gong and Yao [7, 8], Hadjisavvas and Schaible [9], Kimura and Yao [10, 11, 12, 13], Oettli [14], and Zeng et al. [15]. But so far, most papers focused mainly on the existence of solutions and the properties of the solutions, there are few papers which deal with the optimality conditions. Giannessi et al. [16] turned the vector variational inequalities with constraints into another vector variational inequalities without constraints. They gave sufficient conditions for efficient solution and weakly efficient solution of the vector variational inequalities in finite dimensional spaces. Morgan and Romaniello [17] gave scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities in Hilbert space by using the concept of subdifferential of the function. Gong [18] presented the necessary and sufficient conditions for weakly efficient solution, Henig efficient solution, and superefficient solution for the vector equilibrium problems with constraints under the condition of cone-convexity. However, the condition of cone-convexity is too strong. Some generalized convexity has been developed, such as cone-preinvexity (see [19]), cone-convexlikeness (see [20]), cone-subconvexlikeness (see [21]), and generalized cone-convexlikeness (see [22]). Among them, the generalized cone-subconvexlikeness has received more attention. Then, it is important to give the optimality conditions for the solution of (VEPC) under conditions of generalized convexity. Moreover, it appears that no work has been done on the Kuhn-Tucker condition of solution for (VEPC). This paper is the effort in this direction.

In the paper, we study the optimality conditions for the vector equilibrium problems. Firstly, we present the necessary and sufficient conditions for globally efficient solution of (VEPC) under generalized cone-subconvexlikeness. Secondly, we prove that the Kuhn-Tucker condition for (VEPC) is both necessary and sufficient under the condition of cone-preinvexity. Meanwhile, we obtain the optimality conditions for vector optimization problems with constraints and vector variational inequality problems with constraints in Section 4.

## 2. Preliminaries and Definitions

It is well known that

(i)if Open image in new window , then Open image in new window has a base;

(ii)if Open image in new window is a Hausdorff locally convex space, then Open image in new window if and only if Open image in new window has a base;

(iii)if Open image in new window is a separable normed space and Open image in new window is a pointed closed convex cone, then Open image in new window is nonempty (see [1]).

Remark 2.1.

The positive cone in many common Banach spaces possesses strictly positive functionals. However, this is not always the case (see [23]).

Let Open image in new window be an arbitrary nonempty subset and Open image in new window . The symbol Open image in new window denotes the closure of Open image in new window , and Open image in new window denotes the generated cone of Open image in new window , that is, Open image in new window . When Open image in new window is a convex, so is Open image in new window .

Remark 2.2.

Obviously, we have

(i) Open image in new window ;

(ii) Open image in new window ;

(iii)if Open image in new window satisfying for all Open image in new window , Open image in new window , then Open image in new window .

Several definitions of generalized convex mapping have been introduced in literature.

(2)Let Open image in new window be a nonempty subset and let Open image in new window be a convex cone.

A nonempty subset Open image in new window is called invex with respect to Open image in new window , if there exists a mapping Open image in new window such that for any Open image in new window , and Open image in new window , Open image in new window .

- (i)
From [21], we know that Open image in new window is Open image in new window -convexlike on Open image in new window if and only if Open image in new window is a convex set and Open image in new window is Open image in new window -subconvexlike on Open image in new window if and only if Open image in new window is a convex set.

- (ii)
If Open image in new window is a convex set, so is Open image in new window . By Lemma Open image in new window of [24], Open image in new window is convex. This shows that Open image in new window -convexlikeness implies Open image in new window -subconvexlikeness. But in general the converse is not true (see [21]).

- (iii)
It is clear that Open image in new window -subconvexlikeness implies generalized Open image in new window -subconvexlikeness. But in general the converse is not true (see [22]).

Remark 2.4.

For Open image in new window , the invex set is a convex set and the Open image in new window -preinvex mapping is a convex mapping. However, there are mappings which are Open image in new window -preinvex but not convex (see [25]).

Yang [26] proved the following Lemma in Banach space; Chen and Rong [27] generalized the result to topological vector space.

Lemma 2.5.

Assume that Open image in new window . Then Open image in new window is generalized Open image in new window -subconvexlike if and only if Open image in new window is convex.

Lemma 2.6.

Assume that (i) Open image in new window is a nonempty subset and Open image in new window is a convex cone with Open image in new window . (ii) Open image in new window is convex. Then Open image in new window is also convex.

Proof.

By Lemma 2.5 and Remark 2.1(iii), we deduce that Open image in new window is a convex set. It is not difficult to prove that Open image in new window is a convex set.

Note that Open image in new window and the closure of a convex set is convex, then Open image in new window is a convex set. The proof is finished.

Lemma 2.7 (see [1]).

If Open image in new window , Open image in new window , then Open image in new window .

Definition 2.8 (see [6]).

Remark 2.9.

Obviously, Open image in new window is a globally efficient solution of (VEPC), then Open image in new window is also a weakly efficient solution of (VEPC). But in general the converse is not true (see [6]).

## 3. Optimality Conditions

Theorem 3.1.

Proof.

Since Open image in new window is convex and Open image in new window is nonempty, then Open image in new window . Note that Open image in new window and (3.13), and we have Open image in new window . With similar proof of Open image in new window , we can prove that Open image in new window .

We need to show that Open image in new window .

On the other hand, since Open image in new window , Open image in new window , by Lemma 2.7, we have Open image in new window , which is a contradiction with (3.15).

Secondly, we show that Open image in new window .

Note that Open image in new window , and there exists Open image in new window such that Open image in new window

By the arbitrariness of Open image in new window , we have Open image in new window .

Lastly, we show that (3.1) and (3.2) hold.

Thus (3.2) holds.

Then (3.1) holds.

which contradicts (3.26). The proof is finished.

Corollary 3.2.

Assume that (i) Open image in new window is invex with respect to Open image in new window ; (ii) Open image in new window and there exists Open image in new window such that Open image in new window ; (iii) Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window , and Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Then Open image in new window is a globally efficient solution of (VEPC) if and only if there exist Open image in new window and Open image in new window such that (3.1) and (3.2) hold.

Proof.

Since Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Then Open image in new window is Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Thus by Theorem 3.1, the conclusion of Corollary 3.2 holds.

Remark 3.3.

Corollary 3.2 generalizes and improves the recent results of Gong (see [18, Theorem 3.3]). Especially, Corollary 3.2 generalizes and improves in the following several aspects.

(1)The condition that the subset Open image in new window is convex is extended to invex.

(2) Open image in new window is Open image in new window -convex in Open image in new window is extended to Open image in new window -preinvex in Open image in new window

(3) Open image in new window is Open image in new window -convex is extended to Open image in new window -preinvex.

Next, we introduce Gateaux derivative of mapping.

Mapping Open image in new window is called Gateaux derivative of Open image in new window at Open image in new window .

The following theorem shows that the Kuhn-Tucker condition for (VEPC) is both necessary and sufficient.

Theorem 3.4.

Proof.

which contradicts (3.29). The proof is finished.

## 4. Application

As interesting applications of the results of Section 3, we obtain the optimality conditions for vector optimization problems and vector variational inequality problems.

Let Open image in new window be the space of all bounded linear mapping from Open image in new window to Open image in new window . We denote by Open image in new window the value of Open image in new window at Open image in new window .

where Open image in new window is a mapping from Open image in new window to Open image in new window .

Definition 4.1 (see [18]).

If Open image in new window , Open image in new window , and if Open image in new window is a globally efficient solution of (VEPC), then Open image in new window is called a globally efficient solution of (VVIC).

Theorem 4.2.

Proof.

then Open image in new window is invex with respect to Open image in new window Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex with respect to Open image in new window , and Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Thus the conditions of Theorem 3.4 are satisfied. Note that Open image in new window , by Theorem 3.4, then the conclusion of Theorem 4.2 holds.

where Open image in new window is a mapping.

Definition 4.3 (see [18]).

If Open image in new window , Open image in new window , and if Open image in new window is a globally efficient solution of (VEPC), then Open image in new window is called a globally efficient solution of (VOP).

Theorem 4.4.

Proof.

then Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex with respect to Open image in new window , and Open image in new window is Gateaux differentiable at Open image in new window and Open image in new window -preinvex on Open image in new window with respect to Open image in new window . Thus the conditions of Theorem 3.4 are satisfied. Note that Open image in new window , by Theorem 3.4, then the conclusion of Theorem 4.4 holds.

By Theorem 3.1, we have the following result.

Theorem 4.5.

## Notes

### Acknowledgment

This work was supported by the National Natural Science Foundation of China (no. 10771228).

### References

- 1.Jahn J:
*Mathematical Vector Optimization in Partially Ordered Linear Spaces, Methods and Procedures in Mathematical Physics*.*Volume 31*. Peter D. Lang, Frankfurt, Germany; 1986:viii+310.Google Scholar - 2.Ansari QH, Oettli W, Schläger D:
**A generalization of vectorial equilibria.***Mathematical Methods of Operations Research*1997,**46**(2):147–152. 10.1007/BF01217687MathSciNetCrossRefMATHGoogle Scholar - 3.Ansari QH, Konnov IV, Yao JC:
**Characterizations of solutions for vector equilibrium problems.***Journal of Optimization Theory and Applications*2002,**113**(3):435–447. 10.1023/A:1015366419163MathSciNetCrossRefMATHGoogle Scholar - 4.Bianchi M, Hadjisavvas N, Schaible S:
**Vector equilibrium problems with generalized monotone bifunctions.***Journal of Optimization Theory and Applications*1997,**92**(3):527–542. 10.1023/A:1022603406244MathSciNetCrossRefMATHGoogle Scholar - 5.Fu J:
**Simultaneous vector variational inequalities and vector implicit complementarity problem.***Journal of Optimization Theory and Applications*1997,**93**(1):141–151. 10.1023/A:1022653918733MathSciNetCrossRefMATHGoogle Scholar - 6.Gong X-H:
**Connectedness of the solution sets and scalarization for vector equilibrium problems.***Journal of Optimization Theory and Applications*2007,**133**(2):151–161. 10.1007/s10957-007-9196-yMathSciNetCrossRefMATHGoogle Scholar - 7.Gong X-H, Yao JC:
**Connectedness of the set of efficient solutions for generalized systems.***Journal of Optimization Theory and Applications*2008,**138**(2):189–196. 10.1007/s10957-008-9378-2MathSciNetCrossRefMATHGoogle Scholar - 8.Gong X-H, Yao JC:
**Lower semicontinuity of the set of efficient solutions for generalized systems.***Journal of Optimization Theory and Applications*2008,**138**(2):197–205. 10.1007/s10957-008-9379-1MathSciNetCrossRefMATHGoogle Scholar - 9.Hadjisavvas N, Schaible S:
**From scalar to vector equilibrium problems in the quasimonotone case.***Journal of Optimization Theory and Applications*1998,**96**(2):297–309. 10.1023/A:1022666014055MathSciNetCrossRefMATHGoogle Scholar - 10.Kimura K, Yao J-C:
**Semicontinuity of solution mappings of parametric generalized strong vector equilibrium problems.***Journal of Industrial and Management Optimization*2008,**4**(1):167–181.MathSciNetCrossRefMATHGoogle Scholar - 11.Kimura K, Yao J-C:
**Sensitivity analysis of solution mappings of parametric vector quasi-equilibrium problems.***Journal of Global Optimization*2008,**41**(2):187–202. 10.1007/s10898-007-9210-9MathSciNetCrossRefMATHGoogle Scholar - 12.Kimura K, Yao J-C:
**Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems.***Taiwanese Journal of Mathematics*2008,**12**(9):2233–2268.MathSciNetMATHGoogle Scholar - 13.Kimura K, Yao J-C:
**Sensitivity analysis of vector equilibrium problems.***Taiwanese Journal of Mathematics*2008,**12**(3):649–669.MathSciNetMATHGoogle Scholar - 14.Oettli W:
**A remark on vector-valued equilibria and generalized monotonicity.***Acta Mathematica Vietnamica*1997,**22**(1):213–221.MathSciNetMATHGoogle Scholar - 15.Zeng L-C, Wu S-Y, Yao J-C:
**Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems.***Taiwanese Journal of Mathematics*2006,**10**(6):1497–1514.MathSciNetMATHGoogle Scholar - 16.Giannessi F, Mastroeni G, Pellegrini L:
**On the theory of vector optimization and variational inequalities. Image space analysis and separation.**In*Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Nonconvex Optimization and Its Applications*.*Volume 38*. Edited by: Giannessi F. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:153–215.CrossRefGoogle Scholar - 17.Morgan J, Romaniello M:
**Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities.***Journal of Optimization Theory and Applications*2006,**130**(2):309–316. 10.1007/s10957-006-9104-xMathSciNetCrossRefMATHGoogle Scholar - 18.Gong X-H:
**Optimality conditions for vector equilibrium problems.***Journal of Mathematical Analysis and Applications*2008,**342**(2):1455–1466. 10.1016/j.jmaa.2008.01.026MathSciNetCrossRefMATHGoogle Scholar - 19.Ben-Israel A, Mond B:
**What is invexity?***Journal of the Australian Mathematical Society. Series B*1986,**28**(1):1–9. 10.1017/S0334270000005142MathSciNetCrossRefMATHGoogle Scholar - 20.Hayashi M, Komiya H:
**Perfect duality for convexlike programs.***Journal of Optimization Theory and Applications*1982,**38**(2):179–189. 10.1007/BF00934081MathSciNetCrossRefMATHGoogle Scholar - 21.Jeyakumar V:
**A generalization of a minimax theorem of Fan via a theorem of the alternative.***Journal of Optimization Theory and Applications*1986,**48**(3):525–533. 10.1007/BF00940575MathSciNetCrossRefMATHGoogle Scholar - 22.Yang XM:
**Alternative theorems and optimality conditions with weakened convexity.***Opsearch*1992,**29:**125–135.MATHGoogle Scholar - 23.Truong XDH:
**Cones admitting strictly positive functionals and scalarization of some vector optimization problems.***Journal of Optimization Theory and Applications*1997,**93**(2):355–372. 10.1023/A:1022654107386MathSciNetCrossRefMATHGoogle Scholar - 24.Breckner WW, Kassay G:
**A systematization of convexity concepts for sets and functions.***Journal of Convex Analysis*1997,**4**(1):109–127.MathSciNetMATHGoogle Scholar - 25.Weir T, Jeyakumar V:
**A class of nonconvex functions and mathematical programming.***Bulletin of the Australian Mathematical Society*1988,**38**(2):177–189. 10.1017/S0004972700027441MathSciNetCrossRefMATHGoogle Scholar - 26.Yang XM:
**Generalized subconvexlike functions and multiple objective optimization.***Systems Science and Mathematical Sciences*1995,**8**(3):254–259.MathSciNetMATHGoogle Scholar - 27.Chen GY, Rong WD:
**Characterizations of the Benson proper efficiency for nonconvex vector optimization.***Journal of Optimization Theory and Applications*1998,**98**(2):365–384. 10.1023/A:1022689517921MathSciNetCrossRefMATHGoogle Scholar

## Copyright information

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.