Necessary optimality conditions of a D.C. set-valued bilevel optimization problem

Abstract

In this article, we consider a bilevel vector optimization problem where objective and constraints are set valued maps. Our approach consists of using a support function [1–3,14,15,32] together with the convex separation principle for the study of necessary optimality conditions for D.C. bilevel set-valued optimization problems. We give optimality conditions in terms of the strong subdifferential of a cone-convex set-valued mapping introduced by Baier and Jahn 6 and the weak subdifferential of a cone-convex set-valued mapping of Sawaragi and Tanino 28. The bilevel set-valued problem is transformed into a one level set-valued optimization problem using a transformation originated by Ye and Zhu 34. An example illustrating the usefulness of our result is also given.

Keywords:

• Bilevel optimization
• Karush–Kuhn–Tucker multipliers
• necessary optimality conditions
• separation principle
• set-valued mappings
• sub-differential
• weak subdifferential
• support function

Acknowlegements

This work has been done during the N. Gadhi's stay at the TU Bergakademie Freiberg as a research fellow of the Alexander von Humboldt foundation.