Vector Equilibrium Problems Research Articles

Stability of solution mapping for parametric symmetric vector equilibrium problems

This paper is concerned with the stability for a parametric symmetric vector equilibrium problem. A parametric gap function for the parametric symmetric vector equilibrium problem is introduced and investigated. By virtue of this function, we establish the sufficient and necessary conditions for the Hausdorff lower semicontinuity of solution mapping to a parametric symmetric vector equilibrium problem. The results presented in this paper generalize and improve the corresponding results in the recent literature.

A note on solution lower semicontinuity for parametric generalized vector equilibrium problems

By using a density technique, sufficient conditions for lower semicontinuity of strong solutions to a parametric generalized vector equilibrium problem are established, where the monotonicity is not necessary. The obtained results are different from the corresponding ones in the literature (Gong and Yao in J. Optim. Theory Appl. 138:197-205, 2008; Gong in J. Optim. Theory Appl. 139:35-46, 2008; Chen and Li in Pac. J. Optim. 6:141-151, 2010; Li and Fang in J. Optim. Theory Appl. 147:507-515, 2010; Gong and Yao in J. Optim. Theory Appl. 138:189-196, 2008). Some examples are given to illustrate the results.MSC:49K40, 90C29, 90C31.

Weak and strong mixed vector equilibrium problems on non-compact domain

In this paper, we consider two mixed vector equilibrium problems i.e., a weak mixed vector equilibrium problem and a strong mixed vector equilibrium problem which are combinations of a vector equilibrium problem and a vector variational inequality problem. We prove existence results for both the problems in non-compact setting.

Erratum to: Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem

Erratum to: Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem

Existence results for generalized vector equilibrium problems with applications

By a coincidence theorem, some existence theorems of solutions are proved for four types of generalized vector equilibrium problems with moving cones. Applications to the generalized semi-infinite programs with the generalized vector equilibrium constraints under the mild conditions are also given. The results of this paper unify and improve the corresponding results in the previous literature.

A viscosity Cesàro mean approximation method for split generalized vector equilibrium problem and fixed point problem

Abstract In this paper, we introduce and study an explicit iterative method to approximate a common solution of split generalized vector equilibrium problem and fixed point problem for a finite family of nonexpansive mappings in real Hilbert spaces using the viscosity Cesaro mean approximation. We prove a strong convergence theorem for the sequences generated by the proposed iterative scheme. Further we give a numerical example to justify our main result. The results presented in this paper generalize, improve and unify the previously known results in this area.

Derivatives of Set-Valued Maps and Gap Functions for Vector Equilibrium Problems

This paper deals with the set-valued gap functions for vector equilibrium problems and investigates their differential properties using Hadamard directional differentials. Also, contingent and adjacent derivatives of a class of set-valued maps are characterized. Moreover, some basic properties of Φ-contingent and Φ-adjacent cones are given in the presence of a nonsmooth kernel function.

Ekeland's principle for set-valued vector equilibrium problems

In this paper, we introduce a concept of quasi C-lower semicontinuity for set-valued mapping and provide a vector version of Ekeland's theorem related to set-valued vector equilibrium problems. As applications, we derive an existence theorem of weakly efficient solution for set-valued vector equilibrium problems without the assumption of convexity of the constraint set and the assumptions of convexity and monotonicity of the set-valued mapping. We also obtain an existence theorem of ɛ-approximate solution for set-valued vector equilibrium problems without the assumptions of compactness and convexity of the constraint set.

A projection iterative algorithm for strong vector equilibrium problem

In this paper, iterative algorithm for strong vector equilibrium problem (SVEP) is studied. Firstly, an auxiliary problem for SVEP is introduced and the relationships between these two problems are discussed. Then, based on the auxiliary problem, a projection iterative algorithm for SVEP is proposed. Moreover, analysis of convergence of this iterative algorithm is investigated under suitable conditions of continuity and convexity. The main result obtained in this paper generalizes and improves the corresponding ones of Iusem and Sosa [Iusem AN, Sosa W. Iterative algorithms for equilibrium problems. Optimization. 2003;52(3):301–316.] and Cheng and Liu [Cheng B, Liu SY. An iterative algorithm for vector equilibrium problems. J. Lanzhou Univ. (Nat. Sci.). 2009;45(5):105–109.].

The uniqueness and well-posedness of vector equilibrium problems with a representation theorem for the solution set

This paper aims to present some uniqueness and well-posedness results for vector equilibrium problems (for short, VEPs). We first construct a complete metric space M consisting of VEPs satisfying some conditions. Using the method of set-valued analysis, we prove that there exists a dense everywhere residual subset Q of M such that each VEP in Q has a unique solution. Moreover, we introduce and obtain the generalized Hadamard well-posedness and generic Hadamard well-posedness of VEPs by considering the perturbations of both vector-valued functions and feasible sets. As an application, we provide a representation theorem for the solution set to each VEP in M.

On parametric implicit vector variational inequality problems

Recently Huang et al. (Math. Comput. Model. 43:1267-1274, 2006)introduced a class of parametric implicit vector equilibrium problems (for shortPIVEP) and they presented some existence results for a solution of PIVEP. Also,they provided two theorems about upper and lower semi-continuity of the solutionset of PIVEP in a locally convex Hausdorff topological vector space. The paperextends the corresponding results obtained in the setting of topological vectorspaces with mild assumptions and removing the notion of locally non-positivenessat a point and lower semi-continuity of the parametric mapping.

Hölder Continuity of Solutions to Parametric Vector Equilibrium Problems with Nonlinear Scalarization

In this article, using the nonlinear scalarization approach by virtue of the nonlinear scalarization function, commonly known as the Gerstewitz function in the theory of vector optimization, Hölder continuity of solution mappings for both set-valued and single-valued cases to parametric vector equilibrium problems is studied. The nonlinear scalarization function is a powerful tool that plays a key role in the proofs, and its main properties (such as sublinearity, continuity, convexity) are fully employed. Especially, its locally and globally Lipschitz properties are provided and the Lipschitz property is first exploited to investigate the Hölder continuity of solutions.

Well-posed generalized vector equilibrium problems

In this paper, we establish the bounded rationality model M for generalized vector equilibrium problems by using a nonlinear scalarization technique. By using the model M, we introduce a new well-posedness concept for generalized vector equilibrium problems, which unifies its Hadamard and Levitin-Polyak well-posedness. Furthermore, sufficient conditions for the well-posedness for generalized vector equilibrium problems are given. As an application, sufficient conditions on the well-posedness for generalized equilibrium problems are obtained.

Sensitivity Analysis for Vector Equilibrium Problems under Functional Perturbations

This article is devoted to sensitivity analysis for vector equilibrium problems under functional perturbations. We show that the solution mapping is upper semicontinuous. Sufficient conditions for lower semicontinuity and Hölder continuity of the solution mapping are established. Finally, we derive some corollaries for special cases of vector equilibrium problems as examples.

Optimality conditions and duality for nonsmooth vector equilibrium problems with constraints

We consider optimality conditions and duality for a general nonsmooth set-valued vector equilibrium problem with inequality constraints. We focus on the -solution, which contains most other concepts, and the firm solution, which is hardly expressed as a -solution. To face high-level nonsmoothness, we employ several notions of contingent variations as generalized derivatives. As relaxed convexity assumptions, some types of arcwise connectedness conditions are imposed. Both necessary and sufficient optimality conditions of orders are investigated for global -solutions and local firm solutions, with consequences for Henig- and Benson-proper solutions. For duality, the Wolfe and Mond-Weir schemes are dealt with, using first-order contingent variations. We discuss weak, strong, direct and converse duality. As illustrative applications, we choose three optimization-related models: a vector minimization problem with inequality constraints, a cone saddle point problem and a variational inequality. Our results are new or improve several existing ones in the literature.

On topological existence theorems and applications to optimization-related problems

In this paper, we establish a continuous selection theorem and use it to derive five equivalent results on the existence of fixed points, sectional points, maximal elements, intersection points and solutions of variational relations, all in topological settings without linear structures. Then, we study the solution existence of a number of optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria, traffic networks, saddle points, constrained minimization, and abstract economies.

Semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem

This paper deals with the stability for a parametric generalized strong vector equilibrium problem. Under new assumptions, which do not contain any information about the solution set and monotonicity, we establish the lower semicontinuity and upper semicontinuity of the solution set to a parametric generalized strong vector equilibrium problem by using a scalarization method and a density result. These results are improve the corresponding ones in recent literature. Some examples are given to illustrate our results.

Extended Vector Equilibrium Problem

In this paper, we consider an extended vector equilibrium problem and prove some existence results in the setting of Hausdorff topological vector spaces and reflexive Banach spaces. Our results extend and improve some known results in the literature. Some examples are given.

LP Well-Posedness for Bilevel Vector Equilibrium and Optimization Problems with Equilibrium Constraints

The purpose of this paper is introduce several types of Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Base on criterion and characterizations for these types of Levitin-Polyak well-posedness we argue on diameters and Kuratowski’s, Hausdorff’s, or Istrǎtescus measures of noncompactness of approximate solution sets under suitable conditions, and we prove the Levitin-Polyak well-posedness for bilevel vector equilibrium and optimization problems with equilibrium constraints. Obtain a gap function for bilevel vector equilibrium problems with equilibrium constraints using the nonlinear scalarization function and consider relations between these types of LP well-posedness for bilevel vector optimization problems with equilibrium constraints and these types of Levitin-Polyak well-posedness for bilevel vector equilibrium problems with equilibrium constraints under suitable conditions; we prove the Levitin-Polyak well-posedness for bilevel equilibrium and optimization problems with equilibrium constraints.

Extended Mixed Vector Equilibrium Problems

We study extended mixed vector equilibrium problems, namely, extended weak mixed vector equilibrium problem and extended strong mixed vector equilibrium problem in Hausdorff topological vector spaces. Using generalized KKM-Fan theorem (Ben-El-Mechaiekh et al.; 2005), some existence results for both problems are proved in noncompact domain.