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Two-step inertial Bregman alternating minimization algorithm for nonconvex and nonsmooth problems

Author

Listed:
  • Jing Zhao

    (Civil Aviation University of China)

  • Qiao-Li Dong

    (Civil Aviation University of China)

  • Michael Th. Rassias

    (Hellenic Military Academy
    Institute for Advanced Study, Program in Interdisciplinary Studies)

  • Fenghui Wang

    (Luoyang Normal University)

Abstract

In this paper, we propose an algorithm combining Bregman alternating minimization algorithm with two-step inertial force for solving a minimization problem composed of two nonsmooth functions with a smooth one in the absence of convexity. For solving nonconvex and nonsmooth problems, we give an abstract convergence theorem for general descent methods satisfying a sufficient decrease assumption, and allowing a relative error tolerance. Our result holds under the assumption that the objective function satisfies the Kurdyka–Łojasiewicz inequality. The proposed algorithm is shown to satisfy the requirements of our abstract convergence theorem. The convergence is obtained provided an appropriate regularization of the objective function satisfies the Kurdyka–Łojasiewicz inequality. Finally, numerical results are reported to show the effectiveness of the proposed algorithm.

Suggested Citation

  • Jing Zhao & Qiao-Li Dong & Michael Th. Rassias & Fenghui Wang, 2022. "Two-step inertial Bregman alternating minimization algorithm for nonconvex and nonsmooth problems," Journal of Global Optimization, Springer, vol. 84(4), pages 941-966, December.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:4:d:10.1007_s10898-022-01176-6
    DOI: 10.1007/s10898-022-01176-6
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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. Min Li & Zhongming Wu, 2019. "Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 535-565, November.
    3. Zhongming Wu & Min Li & David Z. W. Wang & Deren Han, 2017. "A Symmetric Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(06), pages 1-27, December.
    4. Heinz H. Bauschke & Jérôme Bolte & Jiawei Chen & Marc Teboulle & Xianfu Wang, 2019. "On Linear Convergence of Non-Euclidean Gradient Methods without Strong Convexity and Lipschitz Gradient Continuity," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 1068-1087, September.
    5. Xue Gao & Xingju Cai & Deren Han, 2020. "A Gauss–Seidel type inertial proximal alternating linearized minimization for a class of nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 76(4), pages 863-887, April.
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