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A Symmetric Alternating Direction Method of Multipliers for Separable Nonconvex Minimization Problems

Author

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  • Zhongming Wu

    (School of Economics and Management, Southeast University, Nanjing 211189, P. R. China)

  • Min Li

    (School of Management and Engineering, Nanjing University, Nanjing 210093, P. R. China)

  • David Z. W. Wang

    (School of Civil and Environmental Engineering, Nanyang Technological University, Singapore)

  • Deren Han

    (School of Mathematical Sciences, Jiangsu Key Labratory for NSLSCS, Nanjing Normal University, Nanjing 210023, P. R. China)

Abstract

In this paper, we propose a symmetric alternating method of multipliers for minimizing the sum of two nonconvex functions with linear constraints, which contains the classic alternating direction method of multipliers in the algorithm framework. Based on the powerful Kurdyka–Łojasiewicz property, and under some assumptions about the penalty parameter and objective function, we prove that each bounded sequence generated by the proposed method globally converges to a critical point of the augmented Lagrangian function associated with the given problem. Moreover, we report some preliminary numerical results on solving l1/2 regularized sparsity optimization and nonconvex feasibility problems to indicate the feasibility and effectiveness of the proposed method.

Suggested Citation

  • 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.
    • Handle: RePEc:wsi:apjorx:v:34:y:2017:i:06:n:s0217595917500300
    DOI: 10.1142/S0217595917500300
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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. Radu Ioan Boţ & Ernö Robert Csetnek & Szilárd Csaba László, 2016. "An inertial forward–backward algorithm for the minimization of the sum of two nonconvex functions," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 4(1), pages 3-25, February.
    3. Jonathan Eckstein & Michael C. Ferris, 1998. "Operator-Splitting Methods for Monotone Affine Variational Inequalities, with a Parallel Application to Optimal Control," INFORMS Journal on Computing, INFORMS, vol. 10(2), pages 218-235, May.
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    Citations

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    Cited by:

    1. Zehui Jia & Xue Gao & Xingju Cai & Deren Han, 2021. "Local Linear Convergence of the Alternating Direction Method of Multipliers for Nonconvex Separable Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(1), pages 1-25, January.
    2. Kai Tu & Haibin Zhang & Huan Gao & Junkai Feng, 2020. "A hybrid Bregman alternating direction method of multipliers for the linearly constrained difference-of-convex problems," Journal of Global Optimization, Springer, vol. 76(4), pages 665-693, April.
    3. 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.
    4. Wu, Tingting & Ng, Michael K. & Zhao, Xi-Le, 2021. "Sparsity reconstruction using nonconvex TGpV-shearlet regularization and constrained projection," Applied Mathematics and Computation, Elsevier, vol. 410(C).
    5. 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.
    6. Peixuan Li & Yuan Shen & Suhong Jiang & Zehua Liu & Caihua Chen, 2021. "Convergence study on strictly contractive Peaceman–Rachford splitting method for nonseparable convex minimization models with quadratic coupling terms," Computational Optimization and Applications, Springer, vol. 78(1), pages 87-124, January.
    7. 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.

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