Weak and strong stationarity in generalized bilevel programming and bilevel optimal control

Abstract

In this article, we consider a general bilevel programming problem in reflexive Banach spaces with a convex lower level problem. In order to derive necessary optimality conditions for the bilevel problem, it is transferred to a mathematical program with complementarity constraints (MPCC). We introduce a notion of weak stationarity and exploit the concept of strong stationarity for MPCCs in reflexive Banach spaces, recently developed by the second author, and we apply these concepts to the reformulated bilevel programming problem. Constraint qualifications are presented, which ensure that local optimal solutions satisfy the weak and strong stationarity conditions. Finally, we discuss a certain bilevel optimal control problem by means of the developed theory. Its weak and strong stationarity conditions of Pontryagin-type and some controllability assumptions ensuring strong stationarity of any local optimal solution are presented.

Keywords:

• Bilevel programming
• programming in Banach spaces
• mathematical program with complementarity constraints
• stationarity
• bilevel optimal control

AMS Subject Classifications:

• 46N10
• 49K15
• 90C33
• 90C48

Notes

No potential conflict of interest was reported by the authors.