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

Author

Listed:
  • Didier Aussel

    (Université de Perpignan)

  • Anton Svensson

    (Universidad de Chile)

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.

Suggested Citation

  • Didier Aussel & Anton Svensson, 2019. "Is Pessimistic Bilevel Programming a Special Case of a Mathematical Program with Complementarity Constraints?," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 504-520, May.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:2:d:10.1007_s10957-018-01467-7
    DOI: 10.1007/s10957-018-01467-7
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    Citations

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    Cited by:

    1. Aussel, Didier & Brotcorne, Luce & Lepaul, Sébastien & von Niederhäusern, Léonard, 2020. "A trilevel model for best response in energy demand-side management," European Journal of Operational Research, Elsevier, vol. 281(2), pages 299-315.
    2. Beck, Yasmine & Ljubić, Ivana & Schmidt, Martin, 2023. "A survey on bilevel optimization under uncertainty," European Journal of Operational Research, Elsevier, vol. 311(2), pages 401-426.
    3. Gang Yao & Rui Li & Yang Yang, 2023. "An Improved Multi-Objective Optimization and Decision-Making Method on Construction Sites Layout of Prefabricated Buildings," Sustainability, MDPI, vol. 15(7), pages 1-23, April.
    4. Juan Guillermo Garrido & Pedro Pérez-Aros & Emilio Vilches, 2023. "Random Multifunctions as Set Minimizers of Infinitely Many Differentiable Random Functions," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 86-110, July.
    5. Fakhry, Ramy & Hassini, Elkafi & Ezzeldin, Mohamed & El-Dakhakhni, Wael, 2022. "Tri-level mixed-binary linear programming: Solution approaches and application in defending critical infrastructure," European Journal of Operational Research, Elsevier, vol. 298(3), pages 1114-1131.
    6. Elisabetta Allevi & Didier Aussel & Rossana Riccardi & Domenico Scopelliti, 2024. "Single-Leader-Radner-Equilibrium: A New Approach for a Class of Bilevel Problems Under Uncertainty," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 344-370, January.
    7. Elias S. Helou & Sandra A. Santos & Lucas E. A. Simões, 2020. "Analysis of a New Sequential Optimality Condition Applied to Mathematical Programs with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 433-447, May.

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