Global Approximations of Vector Optimization Problems in Terms of Variational Convergence

Global approximations of scalar optimization problems in terms of variational convergence have been studied for several decades. However, there are very few results for vector models in the literature, which are limited to epi-convergence of vector functions, nothing has been obtained for vector bifunctions. This paper is the first attempt to consider approximations of vector optimization in terms of variational convergence of bifunctions. We first consider the vector quasi-equilibrium problem since it encompasses most optimization-related models. Then, we apply the obtained results to the vector Nash quasi-game (vector generalized non-cooperative game), which is one of the most important practical problems in applied mathematics. We approach to the vector problems in question via scalarization. Our scalarization tools are the Tammer and the Hiriart-Urruty scalarization functions. By scalarizing, we define types of vector variational convergence corresponding to some types of scalar variational convergence. We show that under these types of convergence of problems approximating the vector problems under consideration, approximate solutions of the former problems tend to solutions of the latter ones in the sense of set convergence.