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Local Linear Convergence of the Alternating Direction Method of Multipliers for Nonconvex Separable Optimization Problems

Author

Listed:
  • Zehui Jia

    (Nanjing University of Information Science and Technology)

  • Xue Gao

    (Nanjing Normal University)

  • Xingju Cai

    (Nanjing Normal University)

  • Deren Han

    (Nanjing Normal University
    Beihang University)

Abstract

In this paper, we consider the convergence rate of the alternating direction method of multipliers for solving the nonconvex separable optimization problems. Based on the error bound condition, we prove that the sequence generated by the alternating direction method of multipliers converges locally to a critical point of the nonconvex optimization problem in a linear convergence rate, and the corresponding sequence of the augmented Lagrangian function value converges in a linear convergence rate. We illustrate the analysis by applying the alternating direction method of multipliers to solving the nonconvex quadratic programming problems with simplex constraint, and comparing it with some state-of-the-art algorithms, the proximal gradient algorithm, the proximal gradient algorithm with extrapolation, and the fast iterative shrinkage–thresholding algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:188:y:2021:i:1:d:10.1007_s10957-020-01782-y
    DOI: 10.1007/s10957-020-01782-y
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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. Xingju Cai & Deren Han & Xiaoming Yuan, 2017. "On the convergence of the direct extension of ADMM for three-block separable convex minimization models with one strongly convex function," Computational Optimization and Applications, Springer, vol. 66(1), pages 39-73, January.
    3. 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.
    4. Luana E. Gibbons & Donald W. Hearn & Panos M. Pardalos & Motakuri V. Ramana, 1997. "Continuous Characterizations of the Maximum Clique Problem," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 754-768, August.
    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. Bo Jiang & Tianyi Lin & Shiqian Ma & Shuzhong Zhang, 2019. "Structured nonconvex and nonsmooth optimization: algorithms and iteration complexity analysis," Computational Optimization and Applications, Springer, vol. 72(1), pages 115-157, January.
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