Existence of Solutions of Set Quasi-Optimization Problems Involving Minkowski Difference

Abstract

This paper gives verifiable conditions for the existence of solutions of some set quasi-optimization problems involving Minkowski difference, where the objective maps are defined in complete metric spaces. Our proof method is not based on any scalarizing approach, and our existence results are written in terms of the given data of the problems. Several examples are provided. As applications of the main results of this paper, new results on the existence of robust solutions for uncertain multi-objective quasi-optimization problems with set-valued maps are formulated.