Vector Optimization with Variable Domination Structures

Abstract

Vector optimization with variable domination structures is a growing up and expanding field of applied mathematics that deals with optimization problems where the domination structure is given by a set-valued map acting between abstract or finite-dimensional spaces. Interesting and important applications of vector optimization with variable domination structure arise in economics, psychology, capability of human behavior, in portfolio management, location theory and radiotherapy treatment in medicine. We give a detailed discussion of solution concepts for problems with variable domination structures based on the (pre-) domination structures. We present certain modifications of translation invariant functionals and show characterizations of solutions to vector optimization problems with variable domination structure by means of translation invariant functionals as well as corresponding modifications. Furthermore, we introduce several concepts for approximate solutions to vector optimization problems with variable domination structures and show corresponding characterizations by means of translation invariant functionals. These results are very useful for further research in the field of vector optimization with variable domination structure, especially for deriving optimality conditions, duality assertions and numerical procedures.