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A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces

Author

Listed:
  • Konstantin Sonntag

    (Paderborn University)

  • Bennet Gebken

    (Paderborn University)

  • Georg Müller

    (Heidelberg University)

  • Sebastian Peitz

    (Paderborn University)

  • Stefan Volkwein

    (Konstanz University)

Abstract

The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from Gebken and Peitz (J Optim Theory Appl 188:696–723, 2021) is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the Clarke subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.

Suggested Citation

  • Konstantin Sonntag & Bennet Gebken & Georg Müller & Sebastian Peitz & Stefan Volkwein, 2024. "A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 455-487, October.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02520-4
    DOI: 10.1007/s10957-024-02520-4
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    References listed on IDEAS

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    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. Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Introduction to Nonsmooth Optimization," Springer Books, Springer, edition 127, number 978-3-319-08114-4, April.
    3. Marco Bernreuther & Georg Müller & Stefan Volkwein, 2022. "Efficient scalarization in multiobjective optimal control of a nonsmooth PDE," Computational Optimization and Applications, Springer, vol. 83(2), pages 435-464, November.
    4. Jörg Fliege & Benar Fux Svaiter, 2000. "Steepest descent methods for multicriteria optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(3), pages 479-494, August.
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