SCALARIZATION METHODS FOR MINTY-TYPE VECTOR VARIATIONAL INEQUALITIES

Abstract

Abstract. Many kinds of Minty’s lemmas show that Minty-type varia-tional inequality problems are very closely related to Stampacchia-typevariational inequality problems. Particularly, Minty-type vector varia-tional inequality problems are deeply connected with vector optimizationproblems. Liu et al. [10] considered vector variational inequalities for set-valued mappings by using scalarization approaches considered by Kon-nov [8]. Lee et al. [9] considered two kinds of Stampacchia-type vectorvariational inequalities by using four kinds of Stampacchia-type scalarvariational inequalities and obtain the relations of the solution sets be-tween the six variational inequalities, which are more generalized resultsthan those considered in [10]. In this paper, the author considers theMinty-type case corresponding to the Stampacchia-type case consideredin [9]. 1. Introduction and PreliminariesRecently, there have been usually traditional concentrations on scalarizationapproaches [3, 4, 6, 8-11] which enable us to replace the vector problems un-der consideration with equivalent scalar problems in studying vector problemsincluding vector optimization problems, vector variational inequality problemsand vector equilibrium problems.In particular, Slavov [11] discussed some scalarization techniques and oneapplication of multi-objective optimization problems into a mathematical eco-nomics.In 2009, Jimenez et al. [6] developed a scalarization method in order toobtain scalar versions of their results on the necessary and sucient condi-tions for strict minimizers of a general vector optimization problem, through avariational approach.Konnov [8] also considered a scalarization approach to connect vector vari-ational inequalities into an equivalent scalar variational inequalities with a