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Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?

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Abstract

One of the most commonly used methods for solving bilevel programming problems (whose lower level problem is convex) starts with reformulating it as a mathematical program with complementarity constraints. This is done by replacing the lower level problem by its Karush–Kuhn–Tucker optimality conditions. The obtained mathematical program with complementarity constraints is (locally) solved, but the question of whether a solution of the reformulation yields a solution of the initial bilevel problem naturally arises. The question was first formulated and answered negatively, in a recent work of Dempe and Dutta, for the so-called optimistic approach. We study this question for the pessimistic approach also in the case of a convex lower level problem with a similar answer. Some new notions of local solutions are defined for these minimax-type problems, for which the relations are shown. Some simple counterexamples are given.

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Acknowledgements

This research benefited from the support of the “FMJH Program Gaspard Monge in optimization and operation research”, and from the support to this program from EDF.

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Authors and Affiliations

  1. Université de Perpignan, Perpignan, France

    Didier Aussel

  2. Universidad de Chile, Santiago, Chile

    Anton Svensson

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  1. Didier Aussel
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  2. Anton Svensson
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Correspondence to Didier Aussel.

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Communicated by René Henrion.

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Aussel, D., Svensson, A. Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?. J Optim Theory Appl 181, 504–520 (2019). https://doi.org/10.1007/s10957-018-01467-7

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  • DOI: https://doi.org/10.1007/s10957-018-01467-7

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