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Proximal-like incremental aggregated gradient method with Bregman distance in weakly convex optimization problems

Author

Listed:
  • Zehui Jia

    (Nanjing University of Information Science and Technology)

  • Jieru Huang

    (Nanjing University of Information Science and Technology)

  • Xingju Cai

    (Nanjing Normal University)

Abstract

We focus on a special nonconvex and nonsmooth composite function, which is the sum of the smooth weakly convex component functions and a proper lower semi-continuous weakly convex function. An algorithm called the proximal-like incremental aggregated gradient (PLIAG) method proposed in Zhang et al. (Math Oper Res 46(1): 61–81, 2021) is proved to be convergent and highly efficient to solve convex minimization problems. This algorithm can not only avoid evaluating the exact full gradient which can be expensive in big data models but also weaken the stringent global Lipschitz gradient continuity assumption on the smooth part of the problem. However, under the nonconvex case, there is few analysis on the convergence of the PLIAG method. In this paper, we prove that the limit point of the sequence generated by the PLIAG method is the critical point of the weakly convex problems. Under further assumption that the objective function satisfies the Kurdyka–Łojasiewicz (KL) property, we prove that the generated sequence converges globally to a critical point of the problem. Additionally, we give the convergence rate when the Łojasiewicz exponent is known.

Suggested Citation

  • Zehui Jia & Jieru Huang & Xingju Cai, 2021. "Proximal-like incremental aggregated gradient method with Bregman distance in weakly convex optimization problems," Journal of Global Optimization, Springer, vol. 80(4), pages 841-864, August.
    • Handle: RePEc:spr:jglopt:v:80:y:2021:i:4:d:10.1007_s10898-021-01044-9
    DOI: 10.1007/s10898-021-01044-9
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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. 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. Wei Peng & Hui Zhang & Xiaoya Zhang, 2019. "Nonconvex Proximal Incremental Aggregated Gradient Method with Linear Convergence," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 230-245, October.
    4. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
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    Cited by:

    1. Xiaoya Zhang & Wei Peng & Hui Zhang, 2022. "Inertial proximal incremental aggregated gradient method with linear convergence guarantees," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(2), pages 187-213, October.

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