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Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates

Author

Listed:
  • Pierre Frankel

    (Université Montpellier 2)

  • Guillaume Garrigos

    (Université Montpellier 2
    Universidad Técnica Federico Santa María)

  • Juan Peypouquet

    (Universidad Técnica Federico Santa María)

Abstract

We study the convergence of general descent methods applied to a lower semi-continuous and nonconvex function, which satisfies the Kurdyka–Łojasiewicz inequality in a Hilbert space. We prove that any precompact sequence converges to a critical point of the function, and obtain new convergence rates both for the values and the iterates. The analysis covers alternating versions of the forward–backward method with variable metric and relative errors. As an example, a nonsmooth and nonconvex version of the Levenberg–Marquardt algorithm is detailed.

Suggested Citation

  • Pierre Frankel & Guillaume Garrigos & Juan Peypouquet, 2015. "Splitting Methods with Variable Metric for Kurdyka–Łojasiewicz Functions and General Convergence Rates," Journal of Optimization Theory and Applications, Springer, vol. 165(3), pages 874-900, June.
    • Handle: RePEc:spr:joptap:v:165:y:2015:i:3:d:10.1007_s10957-014-0642-3
    DOI: 10.1007/s10957-014-0642-3
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Dominikus Noll, 2014. "Convergence of Non-smooth Descent Methods Using the Kurdyka–Łojasiewicz Inequality," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 553-572, February.
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    Cited by:

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    14. Radu Ioan Boţ & Ernö Robert Csetnek & Szilárd Csaba László, 2016. "An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 3-25, February.
    15. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "A Block Inertial Bregman Proximal Algorithm for Nonsmooth Nonconvex Problems with Application to Symmetric Nonnegative Matrix Tri-Factorization," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 234-258, July.
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