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Bilevel Linear Optimization Under Uncertainty

In: Bilevel Optimization

Author

Listed:
  • Johanna Burtscheidt

    (University of Duisburg-Essen)

  • Matthias Claus

    (University of Duisburg-Essen)

Abstract

We consider bilevel linear problems, where the right-hand side of the lower level problems is stochastic. The leader has to decide in a here-and-now fashion, while the follower has complete information. In this setting, the leader’s outcome can be modeled by a random variable, which gives rise to a broad spectrum of models involving coherent or convex risk measures and stochastic dominance constraints. We outline Lipschitzian properties, conditions for existence and optimality, as well as stability results. Moreover, for finite discrete distributions, we discuss the special structure of equivalent deterministic bilevel programs and its potential use to mitigate the curse of dimensionality.

Suggested Citation

  • Johanna Burtscheidt & Matthias Claus, 2020. "Bilevel Linear Optimization Under Uncertainty," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 485-511, Springer.
    • Handle: RePEc:spr:spochp:978-3-030-52119-6_17
    DOI: 10.1007/978-3-030-52119-6_17
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    Citations

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    Cited by:

    1. Gianfranco Liberona & David Salas & Léonard Niederhäusern, 2024. "The Value of Shared Information for allocation of drivers in ride-hailing: a proof-of-concept study," Computational Management Science, Springer, vol. 21(1), pages 1-32, June.
    2. Holger Heitsch & René Henrion & Thomas Kleinert & Martin Schmidt, 2022. "On convex lower-level black-box constraints in bilevel optimization with an application to gas market models with chance constraints," Journal of Global Optimization, Springer, vol. 84(3), pages 651-685, November.
    3. 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.
    4. Akshit Goyal & Yiling Zhang & Chuan He, 2023. "Decision Rule Approaches for Pessimistic Bilevel Linear Programs Under Moment Ambiguity with Facility Location Applications," INFORMS Journal on Computing, INFORMS, vol. 35(6), pages 1342-1360, November.
    5. Massimiliano Caramia, 2024. "An algorithm to find stable solutions in linear–linear bilevel problems," OPSEARCH, Springer;Operational Research Society of India, vol. 61(2), pages 972-988, June.
    6. Christoph Buchheim & Dorothee Henke & Jannik Irmai, 2022. "The Stochastic Bilevel Continuous Knapsack Problem with Uncertain Follower’s Objective," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 521-542, August.
    7. 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.
    8. Yasmine Beck & Daniel Bienstock & Martin Schmidt & Johannes Thürauf, 2023. "On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 428-447, July.
    9. Puming Wang & Liqin Zheng & Tianyi Diao & Shengquan Huang & Xiaoqing Bai, 2023. "Robust Bilevel Optimal Dispatch of Park Integrated Energy System Considering Renewable Energy Uncertainty," Energies, MDPI, vol. 16(21), pages 1-23, October.

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