IDEAS home Printed from https://ideas.repec.org/a/spr/opsear/v61y2024i2d10.1007_s12597-023-00735-z.html
   My bibliography  Save this article

An algorithm to find stable solutions in linear–linear bilevel problems

Author

Listed:
  • Massimiliano Caramia

    (Tor Vergata University of Rome)

Abstract

Bilevel programming is widely used to model applications with two hierarchical decisions defining, respectively, an upper-level (leader) problem and a lower-level (follower) problem nested in the former. In this scenario, the stability of the solution is one of the main issues to address to have a well-posed optimization problem; indeed, the lower-level problem may admit more than one optimal solution generating uncertainty on what the real upper-level objective value will be (instability of the bilevel problem). Notwithstanding the importance of stability, the literature has concentrated the most on defining hypotheses and conditions warranting stability of bilevel problems rather than on algorithms able to find a stable solution to the latter. To give a contribution in this direction, in this paper, we define an algorithm capable of finding a stable solution in bilevel optimization problems. We restrict our analysis to the case in which both the leader and the follower problems are linear-continuous. The proposal starts with finding the optimistic solution of the single-level problem obtained by replacing the follower’s problem with its Karush–Kuhn–Tucker conditions, and the corresponding pessimistic solution. In the presence of instability, the algorithm removes one of the follower constraints at a time producing additional constraints for the leader problem in the attempt to define a new stable bilevel problem. Experimental results show that the algorithm can produce stable bilevel solutions representing an effective trade-off between the optimistic and the pessimistic solution values. The proposed algorithm may represent an effective tool for decision-makers aiming at finding robust solutions to bilevel problems.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:opsear:v:61:y:2024:i:2:d:10.1007_s12597-023-00735-z
    DOI: 10.1007/s12597-023-00735-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s12597-023-00735-z
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s12597-023-00735-z?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Hamiden Abd El-Wahed Khalifa & Pavan Kumar, 2023. "Multi-objective optimisation for solving cooperative continuous static games using Karush-Kuhn-Tucker conditions," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 46(1), pages 133-147.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. Miguel F. Arevalo-Castiblanco & Jaime Pachon & Duvan Tellez-Castro & Eduardo Mojica-Nava, 2023. "Cooperative Cruise Control for Intelligent Connected Vehicles: A Bargaining Game Approach," Sustainability, MDPI, vol. 15(15), pages 1-21, August.
    6. Pavan Kumar & Hamiden Abd El-Wahed Khalifa, 2023. "Enhancing Decomposition Approach for Solving Multi-Objective Dynamic Non-Linear Programming Problems Involving Fuzziness," Mathematics, MDPI, vol. 11(14), pages 1-16, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:opsear:v:61:y:2024:i:2:d:10.1007_s12597-023-00735-z. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.