Set-Valued Optimization Problems

Abstract

Set optimization has become a very important field in optimization theory as well as in various applications, especially in welfare economics, mathematical finance, optimization under uncertainty and medical image processing. The aim of this chapter is to show that it is possible to give characterizations of solutions of set-valued optimization problems by means of translation invariant functionals. First, we introduce set relations and define solution concepts for set-valued optimization problems based on these set relations. Hernández and Rodríguez-Marín introduced a functional that can be considered as an extension of translation invariant functionals related to set-valued optimization. We discuss the properties of this functional, especially boundedness and monotonicity. A characterization of the solutions of set-valued optimization problems via scalarization by means of translation invariant functionals is given. We will see that it is possible to derive a complete characterization of solutions to set-valued optimization problems using translation invariant functionals with very useful properties and nice geometrical interpretations.