Approximation of Weak Efficient Solutions in Vector Optimization

In this paper, a well-known concept of epsilon-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 Painleve-Kuratowski sense, of the epsilon-efficient sets when the precision epsilon tends to zero is the set of weak efficient solutions of the problem. Moreover, several nonlinear scalarization results are derived to characterize the epsilon-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 epsilon-efficient solutions of convex multiobjective problems with inequality constraints via multiplier rules.