On the continuity of set to point maps with applications to Pareto optimization

Abstract

Pareto optimization problems have sets as solution both in the decisions’ space and in the objectives’ space. When the problem depends on a real vector parameter the notion of the stability of the solution must be defined and studied. We introduce in this paper the notions of limit of sequences of sets, open, closed and continous application. Then we determine some relations between the stability of the solutions in the decisions’ space and in the objectives’ space.