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Faster Convergence Rates of Relaxed Peaceman-Rachford and ADMM Under Regularity Assumptions

Author

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  • Damek Davis

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, NY)

  • Wotao Yin

    (Department of Mathematics, University of California, Los Angeles, California)

Abstract

In this paper, we provide a comprehensive convergence rate analysis of the Douglas-Rachford splitting (DRS), Peaceman-Rachford splitting (PRS), and alternating direction method of multipliers (ADMM) algorithms under various regularity assumptions including strong convexity, Lipschitz differentiability, and bounded linear regularity. The main consequence of this work is that relaxed PRS and ADMM automatically adapt to the regularity of the problem and achieve convergence rates that improve upon the (tight) worst-case rates that hold in the absence of such regularity. All of the results are obtained using simple techniques.

Suggested Citation

  • 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.
    • Handle: RePEc:inm:ormoor:v:42:y:2017:i:3:p:783-805
    DOI: 10.1287/moor.2016.0827
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    References listed on IDEAS

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    1. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
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    Cited by:

    1. Hedy Attouch & Zaki Chbani & Jalal Fadili & Hassan Riahi, 2022. "Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 704-736, June.
    2. Pauline Tan, 2018. "Linear Convergence Rates for Variants of the Alternating Direction Method of Multipliers in Smooth Cases," Journal of Optimization Theory and Applications, Springer, vol. 176(2), pages 377-398, February.
    3. Abbaszadehpeivasti, Hadi, 2024. "Performance analysis of optimization methods for machine learning," Other publications TiSEM 3050a62d-1a1f-494e-99ef-7, Tilburg University, School of Economics and Management.
    4. Renato D. C. Monteiro & Chee-Khian Sim, 2018. "Complexity of the relaxed Peaceman–Rachford splitting method for the sum of two maximal strongly monotone operators," Computational Optimization and Applications, Springer, vol. 70(3), pages 763-790, July.
    5. Zamani, Moslem & Abbaszadehpeivasti, Hadi & de Klerk, Etienne, 2024. "The exact worst-case convergence rate of the alternating direction method of multipliers," Other publications TiSEM f30ae9e6-ed19-423f-bd1e-0, Tilburg University, School of Economics and Management.
    6. 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.
    7. Naomi Graham & Hao Hu & Jiyoung Im & Xinxin Li & Henry Wolkowicz, 2022. "A Restricted Dual Peaceman-Rachford Splitting Method for a Strengthened DNN Relaxation for QAP," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2125-2143, July.
    8. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.
    9. Chee-Khian Sim, 2023. "Convergence Rates for the Relaxed Peaceman-Rachford Splitting Method on a Monotone Inclusion Problem," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 298-323, January.

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