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An approximation algorithm for multi-objective optimization problems using a box-coverage

Author

Listed:
  • Gabriele Eichfelder

    (Technische Universität Ilmenau)

  • Leo Warnow

    (Technische Universität Ilmenau)

Abstract

For a continuous multi-objective optimization problem, it is usually not a practical approach to compute all its nondominated points because there are infinitely many of them. For this reason, a typical approach is to compute an approximation of the nondominated set. A common technique for this approach is to generate a polyhedron which contains the nondominated set. However, often these approximations are used for further evaluations. For those applications a polyhedron is a structure that is not easy to handle. In this paper, we introduce an approximation with a simpler structure respecting the natural ordering. In particular, we compute a box-coverage of the nondominated set. To do so, we use an approach that, in general, allows us to update not only one but several boxes whenever a new nondominated point is found. The algorithm is guaranteed to stop with a finite number of boxes, each being sufficiently thin.

Suggested Citation

  • Gabriele Eichfelder & Leo Warnow, 2022. "An approximation algorithm for multi-objective optimization problems using a box-coverage," Journal of Global Optimization, Springer, vol. 83(2), pages 329-357, June.
    • Handle: RePEc:spr:jglopt:v:83:y:2022:i:2:d:10.1007_s10898-021-01109-9
    DOI: 10.1007/s10898-021-01109-9
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    References listed on IDEAS

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    1. Gabriele Eichfelder & Peter Kirst & Laura Meng & Oliver Stein, 2021. "Correction to: A general branch-and-bound framework for continuous global multiobjective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 229-229, May.
    2. S. Ruzika & M. M. Wiecek, 2005. "Approximation Methods in Multiobjective Programming," Journal of Optimization Theory and Applications, Springer, vol. 126(3), pages 473-501, September.
    3. Matthias Ehrgott, 2005. "Multicriteria Optimization," Springer Books, Springer, edition 0, number 978-3-540-27659-3, June.
    4. 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.
    5. Panos M. Pardalos & Antanas Žilinskas & Julius Žilinskas, 2017. "Non-Convex Multi-Objective Optimization," Springer Optimization and Its Applications, Springer, number 978-3-319-61007-8, December.
    6. Klamroth, Kathrin & Lacour, Renaud & Vanderpooten, Daniel, 2015. "On the representation of the search region in multi-objective optimization," European Journal of Operational Research, Elsevier, vol. 245(3), pages 767-778.
    7. Gabriele Eichfelder & Peter Kirst & Laura Meng & Oliver Stein, 2021. "A general branch-and-bound framework for continuous global multiobjective optimization," Journal of Global Optimization, Springer, vol. 80(1), pages 195-227, May.
    8. Andreas Löhne & Birgit Rudloff & Firdevs Ulus, 2014. "Primal and dual approximation algorithms for convex vector optimization problems," Journal of Global Optimization, Springer, vol. 60(4), pages 713-736, December.
    9. Rasmus Bokrantz & Anders Forsgren, 2013. "An Algorithm for Approximating Convex Pareto Surfaces Based on Dual Techniques," INFORMS Journal on Computing, INFORMS, vol. 25(2), pages 377-393, May.
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    Cited by:

    1. Yong Zhao & Wang Chen & Xinmin Yang, 2024. "Adaptive Sampling Stochastic Multigradient Algorithm for Stochastic Multiobjective Optimization," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 215-241, January.
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    4. Gabriele Eichfelder & Oliver Stein & Leo Warnow, 2024. "A Solver for Multiobjective Mixed-Integer Convex and Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1736-1766, November.

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