Optimality Conditions for a Simple Convex Bilevel Programming Problem

S. Dempe (), N. Dinh () and J. Dutta ()
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S. Dempe: TU Bergakademie Freiberg
N. Dinh: International University, Vietnam National University of Ho Chi Minh city
J. Dutta: Indian Institute of Technology

A chapter in Variational Analysis and Generalized Differentiation in Optimization and Control, 2010, pp 149-161 from Springer

Abstract: Abstract The problem to find a best solution within the set of optimal solutions of a convex optimization problem is modeled as a bilevel programming problem. It is shown that regularity conditions like Slater’s constraint qualification are never satisfied for this problem. If the lower-level problem is replaced with its (necessary and sufficient) optimality conditions, it is possible to derive a necessary optimality condition for the resulting problem. An example is used to show that this condition in not sufficient even if the initial problem is a convex one. If the lower-level problem is replaced using its optimal value, it is possible to obtain an optimality condition that is both necessary and sufficient in the convex case.

Date: 2010
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DOI: 10.1007/978-1-4419-0437-9_7

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