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On convex lower-level black-box constraints in bilevel optimization with an application to gas market models with chance constraints

Author

Listed:
  • Holger Heitsch

    (Weierstrass Institute for Applied Analysis and Stochastics)

  • René Henrion

    (Weierstrass Institute for Applied Analysis and Stochastics)

  • Thomas Kleinert

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

  • Martin Schmidt

    (Trier University)

Abstract

Bilevel optimization is an increasingly important tool to model hierarchical decision making. However, the ability of modeling such settings makes bilevel problems hard to solve in theory and practice. In this paper, we add on the general difficulty of this class of problems by further incorporating convex black-box constraints in the lower level. For this setup, we develop a cutting-plane algorithm that computes approximate bilevel-feasible points. We apply this method to a bilevel model of the European gas market in which we use a joint chance constraint to model uncertain loads. Since the chance constraint is not available in closed form, this fits into the black-box setting studied before. For the applied model, we use further problem-specific insights to derive bounds on the objective value of the bilevel problem. By doing so, we are able to show that we solve the application problem to approximate global optimality. In our numerical case study we are thus able to evaluate the welfare sensitivity in dependence of the achieved safety level of uncertain load coverage.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:3:d:10.1007_s10898-022-01161-z
    DOI: 10.1007/s10898-022-01161-z
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    References listed on IDEAS

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    1. Veronika Grimm & Lars Schewe & Martin Schmidt & Gregor Zöttl, 2019. "A multilevel model of the European entry-exit gas market," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(2), pages 223-255, April.
    2. 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.
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    5. 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:

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