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An approximation algorithm for multiobjective mixed-integer convex optimization

Author

Listed:
  • Ina Lammel

    (Fraunhofer Institute for Industrial Mathematics (ITWM))

  • Karl-Heinz Küfer

    (Fraunhofer Institute for Industrial Mathematics (ITWM))

  • Philipp Süss

    (Fraunhofer Institute for Industrial Mathematics (ITWM))

Abstract

In this article we introduce an algorithm that approximates the nondominated sets of multiobjective mixed-integer convex optimization problems. The algorithm constructs an inner and outer approximation of the front exploiting the convexity of the patches for problems with an arbitrary number of criteria. In the algorithm, the problem is decomposed into patches, which are multiobjective convex problems, by fixing the integer assignments. The patch problems are solved using (simplicial) Sandwiching. We identify parts of patches that are dominated by other patches and ensure that these patch parts are not refined further. We prove that the algorithm converges and show a bound on the reduction of the approximation error in the course of the algorithm. We illustrate the behaviour of our algorithm using some numerical examples and compare its performance to an algorithm from literature.

Suggested Citation

  • Ina Lammel & Karl-Heinz Küfer & Philipp Süss, 2024. "An approximation algorithm for multiobjective mixed-integer convex optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 321-350, August.
    • Handle: RePEc:spr:mathme:v:100:y:2024:i:1:d:10.1007_s00186-024-00870-3
    DOI: 10.1007/s00186-024-00870-3
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    References listed on IDEAS

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    1. Kathrin Klamroth & Jørgen Tind & Margaret M. Wiecek, 2003. "Unbiased approximation in multicriteria optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(3), pages 413-437, January.
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    3. Cacchiani, Valentina & D’Ambrosio, Claudia, 2017. "A branch-and-bound based heuristic algorithm for convex multi-objective MINLPs," European Journal of Operational Research, Elsevier, vol. 260(3), pages 920-933.
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    5. Hartikainen, Markus & Miettinen, Kaisa & Klamroth, Kathrin, 2019. "Interactive Nonconvex Pareto Navigator for multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 275(1), pages 238-251.
    6. Gijs Rennen & Edwin R. van Dam & Dick den Hertog, 2011. "Enhancement of Sandwich Algorithms for Approximating Higher-Dimensional Convex Pareto Sets," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 493-517, November.
    7. Matthias Ehrgott, 2005. "Multicriteria Optimization," Springer Books, Springer, edition 0, number 978-3-540-27659-3, December.
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    Cited by:

    1. Carlos Henggeler Antunes & Carlos M. Fonseca & Luís Paquete & Michael Stiglmayr, 2024. "Special issue on exact and approximation methods for mixed-integer multi-objective optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 100(1), pages 1-4, August.

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