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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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    6. J. X. Cruz Neto & P. R. Oliveira & A. Soubeyran & J. C. O. Souza, 2020. "A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem," Annals of Operations Research, Springer, vol. 289(2), pages 313-339, June.
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    8. Maryam Yashtini, 2022. "Convergence and rate analysis of a proximal linearized ADMM for nonconvex nonsmooth optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 913-939, December.
    9. Silvia Bonettini & Peter Ochs & Marco Prato & Simone Rebegoldi, 2023. "An abstract convergence framework with application to inertial inexact forward–backward methods," Computational Optimization and Applications, Springer, vol. 84(2), pages 319-362, March.
    10. S. Bonettini & M. Prato & S. Rebegoldi, 2018. "A block coordinate variable metric linesearch based proximal gradient method," Computational Optimization and Applications, Springer, vol. 71(1), pages 5-52, September.
    11. Franck Iutzeler & Jérôme Malick, 2018. "On the Proximal Gradient Algorithm with Alternated Inertia," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 688-710, March.
    12. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2016. "A block coordinate variable metric forward–backward algorithm," Journal of Global Optimization, Springer, vol. 66(3), pages 457-485, November.
    13. Szilárd Csaba László, 2023. "A Forward–Backward Algorithm With Different Inertial Terms for Structured Non-Convex Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 387-427, July.
    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.
    16. Zehui Jia & Xue Gao & Xingju Cai & Deren Han, 2021. "Local Linear Convergence of the Alternating Direction Method of Multipliers for Nonconvex Separable Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 1-25, January.
    17. Maryam Yashtini, 2021. "Multi-block Nonconvex Nonsmooth Proximal ADMM: Convergence and Rates Under Kurdyka–Łojasiewicz Property," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 966-998, September.
    18. Yaohua Hu & Chong Li & Kaiwen Meng & Xiaoqi Yang, 2021. "Linear convergence of inexact descent method and inexact proximal gradient algorithms for lower-order regularization problems," Journal of Global Optimization, Springer, vol. 79(4), pages 853-883, April.

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