On generalized vector variational-like inequalities

In this paper, we consider a vector version of Minty's lemma and obtain existence theorems for two kinds of variational-like inequality. A fixed point theorem is also discussed.     

On system of generalized vector variational inequalities

In this paper, we introduce a new system of generalized vector variational inequalities with variable preference. This extends the model of system of generalized variational inequalities due to Pang and Konnov independently as well as system of vector equilibrium problems due to Ansari, Schaible and Yao. We establish existence of solutions to the new system under weaker conditions that include a new partial diagonally convexity and a weaker notion than continuity. As applications, we derive existence results for both systems of vector variational-like inequalities and vector optimization problems with variable preference.     

On the existence of solutions to generalized vector variational-like inequalities

In this paper, we consider a vector version of generalized Minty's lemma and obtain existence theorems of solutions for two kinds of vector variational-like inequalities.     

Decomposition of generalized vector variational inequalities

A multicriteria optimization problem is called Pareto reducible if its weakly efficient solutions actually are efficient solutions for the problem itself or for at least one subproblem obtained from it by selecting certain criteria. The aim of this paper is to investigate a similar property within a special class of generalized vector variational inequalities, under appropriate generalized convexity assumptions.     

Existence conditions for generalized vector variational inequalities

The aim of this note is to get new results concerning set-valued vector equilibrium problems, which extend some recent assertions in this field.     

On generalized variational inequalities

We obtain a new version of the minimax inequality of Ky Fan. As an application, an existence result for the generalized variational inequality problem with set-valued mappings defined on noncompact sets in Hausdorff topological vector spaces is given. Also, some existence results for the generalized variational inequality problem for quasimonotone and pseudomonotone mappings are obtained. Dedicated to the memory of T. Rapcsák.     

On vector variational inequalities

In this paper, we study vector variational inequalities. The concept of weaklyC-pseudomonotone operator is introduced. By employing the Fan lemma, we establish several existence results. The new results extend and unify existence results of vector variational inequalities for monotone operators under a Banach space setting. In particular, existence results for the generalized vector complementarity problem with weaklyC-pseudomonotone operators in Banach space are obtained.This research was partially supported by the National Science Council of the Republic of China under Contract NSC 84-2121-M-110-008.     

On vector variational inequalities

In this paper, we introduce a general form of a vector variational inequality and prove the existence of its solutions with and without convexity assumptions.     

Gap functions and error bounds for generalized vector variational inequalities

This paper deals with generalized vector variational inequalities. Without any scalarization approach, the gap functions and their regularized versions for generalized vector variational inequalities are first obtained. Then, in the absence of the projection operator method, some error bounds for generalized vector variational inequalities are established in terms of these regularized gap functions. Further, the results obtained in this paper are more simpler from the computational view.     

Existence of solutions for generalized implicit vector variational-like inequalities

In this paper, we introduce a new class of generalized implicit vector variational-like inequalities in Hausdorff topological vector spaces and Banach spaces which contain implicit vector equilibrium problems, implicit vector variational inequalities and implicit vector complementarity problems as special cases. We derive some new results by using the KKM–Fan theorem, under compact and noncompact assumptions on underlying convex sets.     

On generalized Nash games and variational inequalities

We show that for a large class of problems a generalized Nash equilibrium can be calculated by solving a variational inequality. We analyze what solutions are found by this reduction procedure and hint at possible applications.     

Differential properties and optimality conditions for generalized weak vector variational inequalities

In this paper, we study a generalized weak vector variational inequality, which is a generalization of a weak vector variational inequality and a Minty weak vector variational inequality. By virtue of a contingent derivative and a Φ-contingent cone, we investigate differential properties of a class of set-valued maps and obtain an explicit expression of its contingent derivative. We also establish some necessary optimality conditions for solutions of the generalized weak vector variational inequality, which generalize the corresponding results in the literature. Furthermore, we establish some unified necessary and sufficient optimality conditions for local optimal solutions of the generalized weak vector variational inequality. Simultaneously, we also show that there is no gap between the necessary and sufficient conditions under an appropriate condition.     

Some generalized vector variational inequalities and complementarity problems for multivalued mappings

In this paper, we introduce and study a new class of generalized vector variational inequalities and complementarity problems for multivalued mappings. We prove the existence of solutions for this kind of vector variational inequality and discuss the relations between the solutions of the generalized vector variational inequalities and the solutions of generalized vector complementarity problems in Hausdorff topological vector spaces. Our results extend and improve some results in this field.     

On vector optimization problems and vector variational inequalities using convexificators

In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points.