Concepts for Approximate Solutions of Vector Optimization Problems with Variable Order Structures

Abstract

In this paper, we introduce the concepts for approximately minimal, approximately nondominated solutions, and approximate minimizers of vector optimization problems with respect to variable order structures. In order to describe solution concepts, we use a set-valued map and this map is not necessarily a (pointed, convex) cone-valued map. We illustrate the different concepts for approximate solutions by several examples. Important properties of these three different kinds of approximate solutions of vector optimization problems with respect to variable order structures are discussed. Finally, we give some necessary conditions for approximate solutions of vector optimization problems with variable order structures using a vector-valued variant of Ekeland’s variational principle.