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An alternating structure-adapted Bregman proximal gradient descent algorithm for constrained nonconvex nonsmooth optimization problems and its inertial variant

Author

Listed:
  • Xue Gao

    (Hebei University of Technology)

  • Xingju Cai

    (Nanjing Normal University)

  • Xiangfeng Wang

    (East China Normal University)

  • Deren Han

    (Beihang University)

Abstract

We consider the nonconvex nonsmooth minimization problem over abstract sets, whose objective function is the sum of a proper lower semicontinuous biconvex function of the entire variables and two smooth nonconvex functions of their private variables. Fully exploiting the problem structure, we propose an alternating structure-adapted Bregman proximal (ASABP for short) gradient descent algorithm, where the geometry of the abstract set and the function is captured by employing generalized Bregman function. Under the assumption that the underlying function satisfies the Kurdyka–Łojasiewicz property, we prove that each bounded sequence generated by ASABP globally converges to a critical point. We then adopt an inertial strategy to accelerate the ASABP algorithm (IASABP), and utilize a backtracking line search scheme to find “suitable” step sizes, making the algorithm efficient and robust. The global O(1/K) sublinear convergence rate measured by Bregman distance is also established. Furthermore, to illustrate the potential of ASABP and its inertial version (IASABP), we apply them to solving the Poisson linear inverse problem, and the results are promising.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:87:y:2023:i:1:d:10.1007_s10898-023-01300-0
    DOI: 10.1007/s10898-023-01300-0
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    References listed on IDEAS

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    1. 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.
    2. Hui Zhang & Yu-Hong Dai & Lei Guo & Wei Peng, 2021. "Proximal-Like Incremental Aggregated Gradient Method with Linear Convergence Under Bregman Distance Growth Conditions," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 61-81, February.
    3. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. 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.
    5. Masoud Ahookhosh & Le Thi Khanh Hien & Nicolas Gillis & Panagiotis Patrinos, 2021. "Multi-block Bregman proximal alternating linearized minimization and its application to orthogonal nonnegative matrix factorization," Computational Optimization and Applications, Springer, vol. 79(3), pages 681-715, July.
    6. 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.
    7. 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.
    8. 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.
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