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On the hierarchical structure of Pareto critical sets

Author

Listed:
  • Bennet Gebken

    (Paderborn University)

  • Sebastian Peitz

    (Paderborn University)

  • Michael Dellnitz

    (Paderborn University)

Abstract

In this article we show that the boundary of the Pareto critical set of an unconstrained multiobjective optimization problem (MOP) consists of Pareto critical points of subproblems where only a subset of the set of objective functions is taken into account. If the Pareto critical set is completely described by its boundary (e.g., if we have more objective functions than dimensions in decision space), then this can be used to efficiently solve the MOP by solving a number of MOPs with fewer objective functions. If this is not the case, the results can still give insight into the structure of the Pareto critical set.

Suggested Citation

  • Bennet Gebken & Sebastian Peitz & Michael Dellnitz, 2019. "On the hierarchical structure of Pareto critical sets," Journal of Global Optimization, Springer, vol. 73(4), pages 891-913, April.
    • Handle: RePEc:spr:jglopt:v:73:y:2019:i:4:d:10.1007_s10898-019-00737-6
    DOI: 10.1007/s10898-019-00737-6
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    References listed on IDEAS

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    1. T. J. Lowe & J.-F. Thisse & J. E. Ward & R. E. Wendell, 1984. "On Efficient Solutions to Multiple Objective Mathematical Programs," Management Science, INFORMS, vol. 30(11), pages 1346-1349, November.
    2. E. Miglierina & E. Molho & M. Rocca, 2008. "Critical Points Index for Vector Functions and Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 138(3), pages 479-496, September.
    3. M. Dellnitz & O. Schütze & T. Hestermeyer, 2005. "Covering Pareto Sets by Multilevel Subdivision Techniques," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 113-136, January.
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    Cited by:

    1. Bennet Gebken & Sebastian Peitz, 2021. "An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 696-723, March.
    2. Bennet Gebken & Sebastian Peitz, 2021. "Inverse multiobjective optimization: Inferring decision criteria from data," Journal of Global Optimization, Springer, vol. 80(1), pages 3-29, May.
    3. Bennet Gebken & Katharina Bieker & Sebastian Peitz, 2023. "On the structure of regularization paths for piecewise differentiable regularization terms," Journal of Global Optimization, Springer, vol. 85(3), pages 709-741, March.

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