Conditions and parametric representations of approximate minimal elements of a set through scalarization

Abstract

This work deals with approximate solutions in vector optimization problems. These solutions frequently appear when an iterative algorithm is used to solve a vector optimization problem. We consider a concept of approximate efficiency introduced by Kutateladze and widely used in the literature to study this kind of solutions. Working in the objective space, necessary and sufficient conditions for Kutateladze’s approximate elements of the image set are given through scalarization in such a way that these points are approximate solutions for a scalar optimization problem. To obtain sufficient conditions we use monotone functions. A new concept is then introduced to describe the idea of parametric representation of the approximate efficient set. Finally, through scalarization, characterizations and parametric representations for the set of approximate solutions in convex and nonconvex vector optimization problems are proved.