Approximation of Weak Efficient Solutions in Vector Optimization

Abstract

In this paper, a well-known concept of ε-efficient solution due to Kutateladze is studied, in order to approximate the weak efficient solutions of vector optimization problems. In particular, it is proved that the limit, in the Painlevé-Kuratowski sense, of the ε-efficient sets when the precision ε tends to zero is the set of weak efficient solutions of the problem. Moreover, several nonlinear scalarization results are derived to characterize the ε-efficient solutions in terms of approximate solutions of scalar optimization problems. Finally, the obtained results are applied not only to propose a kind of penalization scheme for Kutateladze’s approximate solutions of a cone constrained convex vector optimization problem but also to characterize ε-efficient solutions of convex multiobjective problems with inequality constraints via multiplier rules.