Optimality Conditions for a Simple Convex Bilevel Programming Problem
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.
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We thank the two anonymous referees whose constructive suggestions have improved the presentation of the chapter. The second author, N. Dinh, expresses his sincere thanks to Vietnam National University, HCM city, and to NAFOSTED, Vietnam, for their support.
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