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A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities

Author

Listed:
  • Julia Grübel

    (Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU)
    Energie Campus Nürnberg)

  • Richard Krug

    (Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU))

  • Martin Schmidt

    (Trier University)

  • Winnifried Wollner

    (Universität Hamburg)

Abstract

We present a novel method for mixed-integer optimization problems with multivariate and Lipschitz continuous nonlinearities. In particular, we do not assume that the nonlinear constraints are explicitly given but that we can only evaluate them and that we know their global Lipschitz constants. The algorithm is a successive linear relaxation method in which we alternate between solving a master problem, which is a mixed-integer linear relaxation of the original problem, and a subproblem, which is designed to tighten the linear relaxation of the next master problem by using the Lipschitz information about the respective functions. By doing so, we follow the ideas of Schmidt, Sirvent, and Wollner (Math Program 178(1):449–483 (2019) and Optim Lett 16(5):1355-1372 (2022)) and improve the tackling of multivariate constraints. Although multivariate nonlinearities obviously increase modeling capabilities, their incorporation also significantly increases the computational burden of the proposed algorithm. We prove the correctness of our method and also derive a worst-case iteration bound. Finally, we show the generality of the addressed problem class and the proposed method by illustrating that both bilevel optimization problems with nonconvex and quadratic lower levels as well as nonlinear and mixed-integer models of gas transport can be tackled by our method. We provide the necessary theory for both applications and briefly illustrate the outcomes of the new method when applied to these two problems.

Suggested Citation

  • Julia Grübel & Richard Krug & Martin Schmidt & Winnifried Wollner, 2023. "A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 1077-1117, September.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:3:d:10.1007_s10957-023-02254-9
    DOI: 10.1007/s10957-023-02254-9
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    References listed on IDEAS

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    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. Christoph Buchheim & Renke Kuhlmann & Christian Meyer, 2018. "Combinatorial optimal control of semilinear elliptic PDEs," Computational Optimization and Applications, Springer, vol. 70(3), pages 641-675, July.
    3. Stephan Dempe, 2020. "Bilevel Optimization: Theory, Algorithms, Applications and a Bibliography," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 581-672, Springer.
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