Semivectorial bilevel programming versus scalar bilevel programming

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.

KEYWORDS:

• Bilevel programming
• existence theory
• multiobjective optimization
• optimal control

AMS SUBJECT CLASSIFICATIONS:

• 49J20
• 49J27
• 90C29
• 90C48

Acknowledgments

The authors would like to thank two anonymous referees for their helpful suggestions and remarks.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Stephan Dempe  http://orcid.org/0000-0001-6344-5152

Patrick Mehlitz  http://orcid.org/0000-0002-9355-850X

Additional information

Funding

This work is partially supported by the Deutsche Forschungsgemeinschaft (DFG) grant Analysis and Solution Methods for Bilevel Optimal Control Problems (grant number DE 650/10-1) within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization).