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Why there is no need to use a big-M in linear bilevel optimization: a computational study of two ready-to-use approaches

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  • Thomas Kleinert

    (Friedrich-Alexander-Universität Erlangen-Nürnberg, Discrete Optimization
    Energie Campus Nürnberg)

  • Martin Schmidt

    (Trier University)

Abstract

Linear bilevel optimization problems have gained increasing attention both in theory as well as in practical applications of Operations Research (OR) during the last years and decades. The latter is mainly due to the ability of this class of problems to model hierarchical decision processes. However, this ability makes bilevel problems also very hard to solve. Since no general-purpose solvers are available, a “best-practice” has developed in the applied OR community, in which not all people want to develop tailored algorithms but “just use” bilevel optimization as a modeling tool for practice. This best-practice is the big-M reformulation of the Karush–Kuhn–Tucker (KKT) conditions of the lower-level problem—an approach that has been shown to be highly problematic by Pineda and Morales (2019). Choosing invalid values for M yields solutions that may be arbitrarily bad. Checking the validity of the big-Ms is however shown to be as hard as solving the original bilevel problem in Kleinert et al. (2019). Nevertheless, due to its appealing simplicity, especially w.r.t. the required implementation effort, this ready-to-use approach still is the most popular method. Until now, there has been a lack of approaches that are competitive both in terms of implementation effort and computational cost. In this note we demonstrate that there is indeed another competitive ready-to-use approach: If the SOS-1 technique is applied to the KKT complementarity conditions, adding the simple additional root-node inequality developed by Kleinert et al. (2020) leads to a competitive performance—without having all the possible theoretical disadvantages of the big-M approach.

Suggested Citation

  • Thomas Kleinert & Martin Schmidt, 2023. "Why there is no need to use a big-M in linear bilevel optimization: a computational study of two ready-to-use approaches," Computational Management Science, Springer, vol. 20(1), pages 1-12, December.
  • Handle: RePEc:spr:comgts:v:20:y:2023:i:1:d:10.1007_s10287-023-00435-5
    DOI: 10.1007/s10287-023-00435-5
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    References listed on IDEAS

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    1. G. Constante-Flores & A. J. Conejo & S. Constante-Flores, 2022. "Solving certain complementarity problems in power markets via convex programming," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 30(3), pages 465-491, October.
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    7. S. Siddiqui & S. Gabriel, 2013. "An SOS1-Based Approach for Solving MPECs with a Natural Gas Market Application," Networks and Spatial Economics, Springer, vol. 13(2), pages 205-227, June.
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    Cited by:

    1. Joaquim Dias Garcia & Guilherme Bodin & Alexandre Street, 2024. "BilevelJuMP.jl: Modeling and Solving Bilevel Optimization Problems in Julia," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 327-335, March.
    2. Jacquet, Quentin & van Ackooij, Wim & Alasseur, Clémence & Gaubert, Stéphane, 2024. "Quadratic regularization of bilevel pricing problems and application to electricity retail markets," European Journal of Operational Research, Elsevier, vol. 313(3), pages 841-857.

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