Approximations of Optimization-Related Problems in Terms of Variational Convergence

Abstract

In this paper, we first adjust the known definition of epi/hypo convergence of bivariate extended-real-valued functions for finite-valued ones defined on product sets. Then, we develop basic variational properties of this type of convergence. Using these results, we consider approximations of optimization-related problems in terms of epi/hypo convergence and related types of convergence, for some selected important cases: equilibrium problems, multiobjective optimization, and Nash equilibria. Our approximation results can be viewed also as contributions to qualitative stability (as the terminology "stability" was also used instead of "approximation" for similar results in the literature).