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Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems

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  • Min Li

    (Nanjing University)

  • Zhongming Wu

    (Southeast University)

Abstract

In this paper, we propose generalized splitting methods for solving a class of nonconvex optimization problems. The new methods are extended from the classic Douglas–Rachford and Peaceman–Rachford splitting methods. The range of the new step-sizes even can be enlarged two times for some special cases. The new methods can also be used to solve convex optimization problems. In particular, for convex problems, we propose more relax conditions on step-sizes and other parameters and prove the global convergence and iteration complexity without any additional assumptions. Under the strong convexity assumption on the objective function, the linear convergence rate can be derived easily.

Suggested Citation

  • Min Li & Zhongming Wu, 2019. "Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 183(2), pages 535-565, November.
    • Handle: RePEc:spr:joptap:v:183:y:2019:i:2:d:10.1007_s10957-019-01564-1
    DOI: 10.1007/s10957-019-01564-1
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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. Min Li & Xiaoming Yuan, 2015. "A Strictly Contractive Peaceman-Rachford Splitting Method with Logarithmic-Quadratic Proximal Regularization for Convex Programming," Mathematics of Operations Research, INFORMS, vol. 40(4), pages 842-858, October.
    3. Zhongming Wu & Min Li, 2019. "General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 129-158, May.
    4. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
    5. Zhongming Wu & Min Li & David Z. W. Wang & Deren Han, 2017. "A Symmetric Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problems," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(06), pages 1-27, December.
    6. Francisco J. Aragón Artacho & Jonathan M. Borwein & Matthew K. Tam, 2016. "Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem," Journal of Global Optimization, Springer, vol. 65(2), pages 309-327, June.
    7. Damek Davis & Wotao Yin, 2017. "Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 783-805, August.
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    Cited by:

    1. Qiao-Li Dong & Songnian He & Michael Th. Rassias, 2021. "General splitting methods with linearization for the split feasibility problem," Journal of Global Optimization, Springer, vol. 79(4), pages 813-836, April.
    2. Chih-Sheng Chuang & Hongjin He & Zhiyuan Zhang, 2022. "A unified Douglas–Rachford algorithm for generalized DC programming," Journal of Global Optimization, Springer, vol. 82(2), pages 331-349, February.
    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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