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Nonconvex Proximal Incremental Aggregated Gradient Method with Linear Convergence

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  • Wei Peng

    (National University of Defense Technology)

  • Hui Zhang

    (National University of Defense Technology)

  • Xiaoya Zhang

    (National University of Defense Technology)

Abstract

In this paper, we study the proximal incremental aggregated gradient algorithm for minimizing the sum of L-smooth nonconvex component functions and a proper closed convex function. By exploiting the L-smooth property and using an error bound condition, we can show that the method still enjoys some desired linear convergence properties, even for nonconvex minimization. Actually, we show that the generated iterative sequence globally converges to the stationary point set. Moreover, we give an explicit computable stepsize threshold to guarantee that both the objective value and iterative sequences are R-linearly convergent.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:1:d:10.1007_s10957-019-01538-3
    DOI: 10.1007/s10957-019-01538-3
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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. Paul Tseng & Sangwoon Yun, 2010. "A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training," Computational Optimization and Applications, Springer, vol. 47(2), pages 179-206, October.
    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. Zhi-Quan Luo & Paul Tseng, 1993. "On the Convergence Rate of Dual Ascent Methods for Linearly Constrained Convex Minimization," Mathematics of Operations Research, INFORMS, vol. 18(4), pages 846-867, November.
    5. Amir Beck & Marc Teboulle, 2006. "A Linearly Convergent Dual-Based Gradient Projection Algorithm for Quadratically Constrained Convex Minimization," Mathematics of Operations Research, INFORMS, vol. 31(2), pages 398-417, May.
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    Cited by:

    1. 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.
    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. 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.
    4. 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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