Fuzzy and Exact Optimality Conditions for a Bilevel Set-Valued Problem via Extremal Principles

In this article, we are concerned with a sequence of two set valued optimization problems in which the feasible region of the first one (the upper-level problem) is determined implicitly by the solution set of the second (the lower-level problem). Since bilevel programming problems are in general nonconvex problems even if the problem data are convex, we use the exact extremal principle and the approximate extremal principle introduced by Mordukhovich [14 B.S. Mordukhovich ( 2001 ). The extremal principle and its applications to optimization and economics . In: Optimization and Related Topics ( A. Rubinov and B. Glover , eds.). Applied Optimization Volumes 47 , Kluwer , Dordrecht , The Netherlands , pp. 343 – 369 . [Google Scholar], 15 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications, Grundlehren Series (Fundamental Principles of Mathematical Sciences), 330 and 331, Springer, Berlin . [Google Scholar]] in order to get optimality conditions for this bilevel problem.