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Level-Set Subdifferential Error Bounds and Linear Convergence of Bregman Proximal Gradient Method

Author

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  • Daoli Zhu

    (Shanghai Jiao Tong University
    The Chinese University of Hong Kong)

  • Sien Deng

    (Northern Illinois University)

  • Minghua Li

    (Chongqing University of Arts and Sciences)

  • Lei Zhao

    (Shanghai Jiao Tong University)

Abstract

In this work, we develop a level-set subdifferential error bound condition with an eye toward convergence rate analysis of a variable Bregman proximal gradient (VBPG) method for a broad class of nonsmooth and nonconvex optimization problems. It is proved that the aforementioned condition guarantees linear convergence of VBPG and is weaker than Kurdyka–Łojasiewicz property, weak metric subregularity, and Bregman proximal error bound. Along the way, we are able to derive a number of verifiable conditions for level-set subdifferential error bounds to hold, and necessary conditions and sufficient conditions for linear convergence relative to a level set for nonsmooth and nonconvex optimization problems. The newly established results not only enable us to show that any accumulation point of the sequence generated by VBPG is at least a critical point of the limiting subdifferential or even a critical point of the proximal subdifferential with a fixed Bregman function in each iteration, but also provide a fresh perspective that allows us to explore inner-connections among many known sufficient conditions for linear convergence of various first-order methods.

Suggested Citation

  • Daoli Zhu & Sien Deng & Minghua Li & Lei Zhao, 2021. "Level-Set Subdifferential Error Bounds and Linear Convergence of Bregman Proximal Gradient Method," Journal of Optimization Theory and Applications, Springer, vol. 189(3), pages 889-918, June.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:3:d:10.1007_s10957-021-01865-4
    DOI: 10.1007/s10957-021-01865-4
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    References listed on IDEAS

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    1. 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.
    2. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2014. "Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 107-132, July.
    3. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    4. 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.
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    Cited by:

    1. Sixuan Bai & Minghua Li & Chengwu Lu & Daoli Zhu & Sien Deng, 2022. "The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 220-245, July.

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