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A note on the convergence of ADMM for linearly constrained convex optimization problems

Author

Listed:
  • Liang Chen

    (Hunan University)

  • Defeng Sun

    (National University of Singapore)

  • Kim-Chuan Toh

    (National University of Singapore)

Abstract

This note serves two purposes. Firstly, we construct a counterexample to show that the statement on the convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained convex optimization problems in a highly influential paper by Boyd et al. (Found Trends Mach Learn 3(1):1–122, 2011) can be false if no prior condition on the existence of solutions to all the subproblems involved is assumed to hold. Secondly, we present fairly mild conditions to guarantee the existence of solutions to all the subproblems of the ADMM and provide a rigorous convergence analysis on the ADMM with a computationally more attractive large step-length that can even exceed the practically much preferred golden ratio of $$(1+\sqrt{5})/2$$ ( 1 + 5 ) / 2 .

Suggested Citation

  • Liang Chen & Defeng Sun & Kim-Chuan Toh, 2017. "A note on the convergence of ADMM for linearly constrained convex optimization problems," Computational Optimization and Applications, Springer, vol. 66(2), pages 327-343, March.
    • Handle: RePEc:spr:coopap:v:66:y:2017:i:2:d:10.1007_s10589-016-9864-7
    DOI: 10.1007/s10589-016-9864-7
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    Cited by:

    1. 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.
    2. Sedi Bartz & Rubén Campoy & Hung M. Phan, 2022. "An Adaptive Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1019-1055, December.
    3. Yao, Yu & Zhu, Xiaoning & Dong, Hongyu & Wu, Shengnan & Wu, Hailong & Carol Tong, Lu & Zhou, Xuesong, 2019. "ADMM-based problem decomposition scheme for vehicle routing problem with time windows," Transportation Research Part B: Methodological, Elsevier, vol. 129(C), pages 156-174.
    4. Deren Han & Defeng Sun & Liwei Zhang, 2018. "Linear Rate Convergence of the Alternating Direction Method of Multipliers for Convex Composite Programming," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 622-637, May.

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