On Optimization Problems with Set‐Valued Objective Maps: Existence and Optimality

In this paper, we consider constrained optimization problems with set‐valued objective maps. First, we define three types of quasi orderings on the set of all non‐empty subsets of n‐dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi‐continuity for set‐valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set‐valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.