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A Gauss–Seidel type inertial proximal alternating linearized minimization for a class of nonconvex optimization problems

Author

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  • Xue Gao

    (Nanjing Normal University)

  • Xingju Cai

    (Nanjing Normal University)

  • Deren Han

    (Beihang University)

Abstract

In this paper we study a broad class of nonconvex and nonsmooth minimization problems, whose objective function is the sum of a smooth function of the entire variables and two nonsmooth functions of each variable. We adopt the framework of the proximal alternating linearized minimization (PALM), together with the inertial strategy to accelerate the convergence. Since the inertial step is performed once the x-subproblem/y-subproblem is updated, the algorithm is a Gauss–Seidel type inertial proximal alternating linearized minimization (GiPALM) algorithm. Under the assumption that the underlying functions satisfy the Kurdyka–Łojasiewicz (KL) property and some suitable conditions on the parameters, we prove that each bounded sequence generated by GiPALM globally converges to a critical point. We apply the algorithm to signal recovery, image denoising and nonnegative matrix factorization models, and compare it with PALM and the inertial proximal alternating linearized minimization.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:4:d:10.1007_s10898-019-00819-5
    DOI: 10.1007/s10898-019-00819-5
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    References listed on IDEAS

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    1. Boţ, Radu Ioan & Csetnek, Ernö Robert & Hendrich, Christopher, 2015. "Inertial Douglas–Rachford splitting for monotone inclusion problems," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 472-487.
    2. 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.
    3. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, September.
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    Cited by:

    1. Xue Gao & Xingju Cai & Xiangfeng Wang & Deren Han, 2023. "An alternating structure-adapted Bregman proximal gradient descent algorithm for constrained nonconvex nonsmooth optimization problems and its inertial variant," Journal of Global Optimization, Springer, vol. 87(1), pages 277-300, September.
    2. Zhongming Wu & Chongshou Li & Min Li & Andrew Lim, 2021. "Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 79(3), pages 617-644, March.
    3. 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.

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