Solution Concepts in Vector Optimization

Abstract

Many applications require the optimization of multiple conflicting goals at the same time. Such a problem can be modeled as a vector optimization problem. Vector optimization deals with the problem of finding efficient elements of a vector-valued function. In that sense, vector optimization generalizes the concept of scalar optimization. In scalar optimization, there is only one concept for efficiency which characterizes efficient elements, namely the solution which generates the smallest function value. But, due to the lack of a total order in general spaces, order relations that are defined within the optimality concept need to be chosen. In this chapter, we discuss several solution concepts for a vector optimization problem. In particular, solution concepts for vector optimization problem equipped with a variable domination structure are studied. Moreover, we present some existence results for solutions of vector optimization problems.