SCALARIZATION METHODS FOR VECTOR VARIATIONAL INEQUALITIES

Abstract

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].