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

Grübel J, Krug R, Schmidt M, Wollner W (2023)

Publication Type: Journal article, Original article

Publication year: 2023

Journal

Journal of Optimization Theory and Applications Springer Verlag (Germany)

Book Volume: 198

Pages Range: 1077-1117

DOI: 10.1007/s10957-023-02254-9

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.

Authors with CRIS profile

Julia Grübel Lehrstuhl für Volkswirtschaftslehre, insbesondere Wirtschaftstheorie Richard Krug Sonderforschungsbereich/Transregio 154: Mathematische Modellierung, Simulation und Optimierung am Beispiel von Gasnetzwerken

Related research project(s)

Multilevel mixed-integer nonlinear optimization for gas markets (B08) (2018 - 2022) TRR 154: Mathematical Modelling, Simulation and Optimisation Using the Example of Gas Networks (TRR 154) July 1, 2018 - June 30, 2022

Involved external institutions

Universität Trier DE Germany (DE) Universität Hamburg (UHH) DE Germany (DE)

How to cite

APA:

Grübel, J., Krug, R., Schmidt, M., & Wollner, W. (2023). A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities. Journal of Optimization Theory and Applications, 198, 1077-1117. https://dx.doi.org/10.1007/s10957-023-02254-9

MLA:

Grübel, Julia, et al. "A Successive Linear Relaxation Method for MINLPs with Multivariate Lipschitz Continuous Nonlinearities." Journal of Optimization Theory and Applications 198 (2023): 1077-1117.

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