Semivectorial bilevel programming versus scalar bilevel programming. (2nd April 2020) - Dempe, Stephan; Mehlitz, Patrick
ABSTRACT: We consider an optimistic semivectorial bilevel programming problem in Banach spaces. The associated lower level multicriteria optimization problem is assumed to be convex w.r.t. its decision variable. This property implies that all its weakly efficient points can be computed applying the weighted-sum-scalarization technique. Consequently, it is possible to replace the overall semivectorial bilevel programming problem by means of a standard bilevel programming problem whose upper level variables comprise the set of suitable scalarization parameters for the lower level problem. In this note, we consider the relationship between this surrogate bilevel programming problem and the original semivectorial bilevel programming problem. As it will be shown, this is a delicate issue as long as locally optimal solutions are investigated. The obtained theory is applied in order to derive existence results for semivectorial bilevel programming problems with not necessarily finite-dimensional lower level decision variables. Some regarding examples from bilevel optimal control are presented.