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On a Computationally Ill-Behaved Bilevel Problem with a Continuous and Nonconvex Lower Level

Author

Listed:
  • Yasmine Beck

    (Trier University)

  • Daniel Bienstock

    (Columbia University)

  • Martin Schmidt

    (Trier University)

  • Johannes Thürauf

    (Trier University)

Abstract

It is well known that bilevel optimization problems are hard to solve both in theory and practice. In this paper, we highlight a further computational difficulty when it comes to solving bilevel problems with continuous but nonconvex lower levels. Even if the lower-level problem is solved to $$\varepsilon $$ ε -feasibility regarding its nonlinear constraints for an arbitrarily small but positive $$\varepsilon $$ ε , the obtained bilevel solution as well as its objective value may be arbitrarily far away from the actual bilevel solution and its actual objective value. This result even holds for bilevel problems for which the nonconvex lower level is uniquely solvable, for which the strict complementarity condition holds, for which the feasible set is convex, and for which Slater’s constraint qualification is satisfied for all feasible upper-level decisions. Since the consideration of $$\varepsilon $$ ε -feasibility cannot be avoided when solving nonconvex problems to global optimality, our result shows that computational bilevel optimization with continuous and nonconvex lower levels needs to be done with great care. Finally, we illustrate that the nonlinearities in the lower level are the key reason for the observed bad behavior by showing that linear bilevel problems behave much better at least on the level of feasible solutions.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:joptap:v:198:y:2023:i:1:d:10.1007_s10957-023-02238-9
    DOI: 10.1007/s10957-023-02238-9
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    References listed on IDEAS

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    1. Christoph Buchheim & Dorothee Henke, 2022. "The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective," Journal of Global Optimization, Springer, vol. 83(4), pages 803-824, August.
    2. R. Paulavičius & C. S. Adjiman, 2020. "New bounding schemes and algorithmic options for the Branch-and-Sandwich algorithm," Journal of Global Optimization, Springer, vol. 77(2), pages 197-225, June.
    3. 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.
    4. Polyxeni-Margarita Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: Theoretical development," Journal of Global Optimization, Springer, vol. 60(3), pages 425-458, November.
    5. Matteo Fischetti & Ivana Ljubić & Michele Monaci & Markus Sinnl, 2017. "A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs," Operations Research, INFORMS, vol. 65(6), pages 1615-1637, December.
    6. Polyxeni-M. Kleniati & Claire Adjiman, 2014. "Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part II: Convergence analysis and numerical results," Journal of Global Optimization, Springer, vol. 60(3), pages 459-481, November.
    7. 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.
    8. Thomas Kleinert & Martine Labbé & Fr¨ank Plein & Martin Schmidt, 2020. "Technical Note—There’s No Free Lunch: On the Hardness of Choosing a Correct Big-M in Bilevel Optimization," Operations Research, INFORMS, vol. 68(6), pages 1716-1721, November.
    9. 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.
    10. Alexander Mitsos, 2010. "Global solution of nonlinear mixed-integer bilevel programs," Journal of Global Optimization, Springer, vol. 47(4), pages 557-582, August.
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