Set-valued Vector Variational Inequalities Research Articles

Ky Fan minimax inequalities for set-valued mappings on dense sets

We give a natural Ky Fan minimax inequality version of set-valued maps, and we deal with a type of vector equilibrium problem for set-valued mappings on a special dense set not on the whole domain. We use these results as applications to study the solutions of a generalized set-valued vector variational inequality.

General set-valued vector variational inequality problems

General set-valued vector variational inequality problems

Degree Theory and Solution Existence of Set-Valued Vector Variational Inequalities in Reflexive Banach Spaces

In this paper, a degree theory for set-valued vector variational inequalities is built in reflexive Banach spaces. By using the method of degree theory, some existence results of solutions for set-valued vector variational inequalities are established under suitable conditions. Furthermore, some equivalent characterizations for the nonemptiness and boundedness of solution sets to single-valued vector variational inequalities are obtained under pseudomonotonicity assumption. To the best of our knowledge, there are still no papers dealing with the degree theory for vector variational inequalities.

Scalarization of set-valued optimization problems and variational inequalities in topological vector spaces

This paper deals with set-valued vector optimization problems and set-valued vector variational inequalities in topological vector spaces, and provides some scalarization approaches for these problems by means of the polar cone and Gerstewitz’s scalarization functions.

Weak sharp minima for set-valued vector variational inequalities with an application

In this paper, the notion of weak sharp minima is employed to the investigation of set-valued vector variational inequalities. The gap function φ T for set-valued strong vector variational inequalities (for short, SVVI) is proved to be less than the gap function ϕ T for set-valued weak vector variational inequalities (for short, WVVI) under certain conditions, which implies that the solution set of SVVI is equivalent to the solution set of WVVI. Moreover, it is shown that weak sharp minima for the solution sets of SVVI and WVVI hold for min 1 ⩽ i ⩽ n p T i and for gap functions φ T and ϕ T under the assumption of strong pseudomonotonicity, where p T i is a gap function for i-th component of SVVI and WVVI. As an application, the weak Pareto solution set of vector optimization problems (for short, VOP) is proved to be weak sharp minimum for min 1 ⩽ i ⩽ n p ∇ g i when each component g i of objective function is strongly convex.

SCALARIZATION METHODS FOR VECTOR VARIATIONAL INEQUALITIES

1. Introduction and preliminariesIn studying vector problems including vector optimization prob-lems, vectorvariationalinequalityproblemsandvectorequilibriumprob-lems, there have been usually traditional concentrations on scalarizationapproaches which enable us to replace the vector problems under con-sideration with equivalent scalar problems [3, 4, 6, 8-10].In particular, in [10], Slavov discussed some scalarization techniquesand one application of the multi-objective optimization problem into amathematical economics.In [6], Jimenez et al. developed a scalarization method in order toobtain scalar versions of their results on the necessary and sufficientconditions for strict minimizers of a general vector optimization problem,through a variational approach.In [8], Konnov considered a scalarization approach to connect vectorvariational inequalities into an equivalent scalar variational inequalitieswith a set-valued cost mapping. He gave an equivalence between weakand strong solutions of set-valued vector variational inequalities andsuggested a new gap function for vector variational inequalities. Heapplied his results to vector optimization, vector network equilibriumand vector migration equilibrium problems.In [4], Guu et al. extended the scalarization approaches of Giannessiet. al [3] to set-valued vector optimization problems and set-valued weakvector optimization problems. The scalar variational inequalities givenby them is different from those given by Konnov [8].

Scalarization approaches for set-valued vector optimization problems and vector variational inequalities

The scalarization approaches of Giannessi, Mastroeni and Pellegrini are extended to the study of set-valued vector optimization problems and set-valued weak vector optimization problems. Some equivalence results among set-valued (scalar) optimization problems, set-valued (scalar) quasi-optimization problems, set-valued vector optimization problems and set-valued weak vector optimization problems are established under the convexity assumption of objective functions. Some examples are provided to illustrate these results. The approaches are furthermore exploited to investigate the set-valued vector variational inequalities and set-valued weak vector variational inequalities, which are different from that suggested by Konnov. Some equivalence relations among set-valued (scalar) variational inequalities, set-valued (scalar) quasi-variational inequalities, set-valued vector variational inequalities and set-valued weak vector variational inequalities are also derived under some suitable conditions.

A Scalarization Approach for Vector Variational Inequalities with Applications

We consider an approach to convert vector variational inequalities into an equivalent scalar variational inequality problem with a set-valued cost mapping. Being based on this property, we give an equivalence result between weak and strong solutions of set-valued vector variational inequalities and suggest a new gap function for vector variational inequalities. Additional examples of applications in vector optimization, vector network equilibrium and vector migration equilibrium problems are also given

Gap Functions and Existence of Solutions to Set-Valued Vector Variational Inequalities

The variational inequality problem with set-valued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.

Weak Sharp Solutions of Variational Inequalities

Weak Sharp Solutions of Variational Inequalities