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A Note on the Alternating Direction Method of Multipliers

Author

Listed:
  • Deren Han

    (Nanjing Normal University)

  • Xiaoming Yuan

    (Hong Kong Baptist University)

Abstract

We consider the linearly constrained separable convex programming, whose objective function is separable into m individual convex functions without coupled variables. The alternating direction method of multipliers has been well studied in the literature for the special case m=2, while it remains open whether its convergence can be extended to the general case m≥3. This note shows the global convergence of this extension when the involved functions are further assumed to be strongly convex.

Suggested Citation

  • Deren Han & Xiaoming Yuan, 2012. "A Note on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 227-238, October.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:1:d:10.1007_s10957-012-0003-z
    DOI: 10.1007/s10957-012-0003-z
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    References listed on IDEAS

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    1. Sun, Jie & Zhang, Su, 2010. "A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs," European Journal of Operational Research, Elsevier, vol. 207(3), pages 1210-1220, December.
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    Cited by:

    1. Liu, Zhiyuan & Zhang, Honggang & Zhang, Kai & Zhou, Zihan, 2023. "Integrating alternating direction method of multipliers and bush for solving the traffic assignment problem," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 177(C).
    2. Yangyang Xu, 2019. "Asynchronous parallel primal–dual block coordinate update methods for affinely constrained convex programs," Computational Optimization and Applications, Springer, vol. 72(1), pages 87-113, January.
    3. Maryam Yashtini, 2022. "Convergence and rate analysis of a proximal linearized ADMM for nonconvex nonsmooth optimization," Journal of Global Optimization, Springer, vol. 84(4), pages 913-939, December.
    4. Puya Latafat & Panagiotis Patrinos, 2017. "Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators," Computational Optimization and Applications, Springer, vol. 68(1), pages 57-93, September.
    5. Yuning Yang & Qingzhi Yang & Su Zhang, 2014. "Modified Alternating Direction Methods for the Modified Multiple-Sets Split Feasibility Problems," Journal of Optimization Theory and Applications, Springer, vol. 163(1), pages 130-147, October.
    6. William W. Hager & Hongchao Zhang, 2019. "Inexact alternating direction methods of multipliers for separable convex optimization," Computational Optimization and Applications, Springer, vol. 73(1), pages 201-235, May.
    7. V. A. Adona & M. L. N. Gonçalves & J. G. Melo, 2019. "Iteration-complexity analysis of a generalized alternating direction method of multipliers," Journal of Global Optimization, Springer, vol. 73(2), pages 331-348, February.
    8. Kaizhao Sun & X. Andy Sun, 2023. "A two-level distributed algorithm for nonconvex constrained optimization," Computational Optimization and Applications, Springer, vol. 84(2), pages 609-649, March.
    9. K. Wang & D. R. Han & L. L. Xu, 2013. "A Parallel Splitting Method for Separable Convex Programs," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 138-158, October.
    10. Wenli Huang & Yuchao Tang & Meng Wen & Haiyang Li, 2022. "Relaxed Variable Metric Primal-Dual Fixed-Point Algorithm with Applications," Mathematics, MDPI, vol. 10(22), pages 1-16, November.
    11. Tahereh Khodamoradi & Maziar Salahi, 2023. "Extended mean-conditional value-at-risk portfolio optimization with PADM and conditional scenario reduction technique," Computational Statistics, Springer, vol. 38(2), pages 1023-1040, June.
    12. Bin Gao & Feng Ma, 2018. "Symmetric Alternating Direction Method with Indefinite Proximal Regularization for Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 178-204, January.
    13. William W. Hager & Hongchao Zhang, 2020. "Convergence rates for an inexact ADMM applied to separable convex optimization," Computational Optimization and Applications, Springer, vol. 77(3), pages 729-754, December.
    14. 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.
    15. Maryam Yashtini, 2021. "Multi-block Nonconvex Nonsmooth Proximal ADMM: Convergence and Rates Under Kurdyka–Łojasiewicz Property," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 966-998, September.
    16. Bingsheng He & Xiaoming Yuan, 2018. "A class of ADMM-based algorithms for three-block separable convex programming," Computational Optimization and Applications, Springer, vol. 70(3), pages 791-826, July.
    17. Ruslan Abdulkadirov & Pavel Lyakhov & Nikolay Nagornov, 2023. "Survey of Optimization Algorithms in Modern Neural Networks," Mathematics, MDPI, vol. 11(11), pages 1-37, May.
    18. Ruoyu Sun & Zhi-Quan Luo & Yinyu Ye, 2020. "On the Efficiency of Random Permutation for ADMM and Coordinate Descent," Mathematics of Operations Research, INFORMS, vol. 45(1), pages 233-271, February.
    19. Bingsheng He & Min Tao & Xiaoming Yuan, 2017. "Convergence Rate Analysis for the Alternating Direction Method of Multipliers with a Substitution Procedure for Separable Convex Programming," Mathematics of Operations Research, INFORMS, vol. 42(3), pages 662-691, August.
    20. Yaning Jiang & Deren Han & Xingju Cai, 2022. "An efficient partial parallel method with scaling step size strategy for three-block convex optimization problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 383-419, December.
    21. Hongsheng Liu & Shu Lu, 2019. "Convergence of the augmented decomposition algorithm," Computational Optimization and Applications, Springer, vol. 72(1), pages 179-213, January.
    22. Liusheng Hou & Hongjin He & Junfeng Yang, 2016. "A partially parallel splitting method for multiple-block separable convex programming with applications to robust PCA," Computational Optimization and Applications, Springer, vol. 63(1), pages 273-303, January.

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