IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v77y2020i3d10.1007_s10589-020-00221-y.html
   My bibliography  Save this article

Convergence rates for an inexact ADMM applied to separable convex optimization

Author

Listed:
  • William W. Hager

    (University of Florida)

  • Hongchao Zhang

    (Louisiana State University)

Abstract

Convergence rates are established for an inexact accelerated alternating direction method of multipliers (I-ADMM) for general separable convex optimization with a linear constraint. Both ergodic and non-ergodic iterates are analyzed. Relative to the iteration number k, the convergence rate is $$\mathcal{{O}}(1/k)$$ O ( 1 / k ) in a convex setting and $$\mathcal{{O}}(1/k^2)$$ O ( 1 / k 2 ) in a strongly convex setting. When an error bound condition holds, the algorithm is 2-step linearly convergent. The I-ADMM is designed so that the accuracy of the inexact iteration preserves the global convergence rates of the exact iteration, leading to better numerical performance in the test problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:77:y:2020:i:3:d:10.1007_s10589-020-00221-y
    DOI: 10.1007/s10589-020-00221-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-020-00221-y
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10589-020-00221-y?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. 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.
    2. Min Li & Defeng Sun & Kim-Chuan Toh, 2015. "A Convergent 3-Block Semi-Proximal ADMM for Convex Minimization Problems with One Strongly Convex Block," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 32(04), pages 1-19.
    3. Caihua Chen & Yuan Shen & Yanfei You, 2013. "On the Convergence Analysis of the Alternating Direction Method of Multipliers with Three Blocks," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-7, October.
    4. Yunmei Chen & William Hager & Maryam Yashtini & Xiaojing Ye & Hongchao Zhang, 2013. "Bregman operator splitting with variable stepsize for total variation image reconstruction," Computational Optimization and Applications, Springer, vol. 54(2), pages 317-342, March.
    5. Bingsheng He & Feng Ma & Xiaoming Yuan, 2020. "Optimally linearizing the alternating direction method of multipliers for convex programming," Computational Optimization and Applications, Springer, vol. 75(2), pages 361-388, March.
    6. D.R. Han & H.K. Lo, 2002. "New Alternating Direction Method for a Class of Nonlinear Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 112(3), pages 549-560, March.
    7. 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.
    8. M. V. Solodov & B. F. Svaiter, 2000. "An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 214-230, May.
    9. Min Li & Li-Zhi Liao & Xiaoming Yuan, 2013. "Inexact Alternating Direction Methods of Multipliers with Logarithmic–Quadratic Proximal Regularization," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 412-436, November.
    10. 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.
    11. 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.
    12. Jonathan Eckstein & Wang Yao, 2017. "Approximate ADMM algorithms derived from Lagrangian splitting," Computational Optimization and Applications, Springer, vol. 68(2), pages 363-405, November.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hélène Le Cadre & Yuting Mou & Hanspeter Höschle, 2020. "Parametrized Inexact-ADMM to Span the Set of Generalized Nash Equilibria: A Normalized Equilibrium Approach," Working Papers hal-02925005, HAL.
    2. Jianchao Bai & William W. Hager & Hongchao Zhang, 2022. "An inexact accelerated stochastic ADMM for separable convex optimization," Computational Optimization and Applications, Springer, vol. 81(2), pages 479-518, March.
    3. Shengjie Xu & Bingsheng He, 2021. "A parallel splitting ALM-based algorithm for separable convex programming," Computational Optimization and Applications, Springer, vol. 80(3), pages 831-851, December.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Jianchao Bai & William W. Hager & Hongchao Zhang, 2022. "An inexact accelerated stochastic ADMM for separable convex optimization," Computational Optimization and Applications, Springer, vol. 81(2), pages 479-518, March.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
    7. Jiaxin Xie, 2018. "On inexact ADMMs with relative error criteria," Computational Optimization and Applications, Springer, vol. 71(3), pages 743-765, December.
    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. 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.
    10. 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.
    11. 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).
    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. Zamani, Moslem & Abbaszadehpeivasti, Hadi & de Klerk, Etienne, 2023. "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.
    14. 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.
    15. Lo, Hong K. & Szeto, W.Y., 2005. "Road pricing modeling for hyper-congestion," Transportation Research Part A: Policy and Practice, Elsevier, vol. 39(7-9), pages 705-722.
    16. Xiaolong Qin & Shin Kang & Yeol Cho, 2010. "Approximating zeros of monotone operators by proximal point algorithms," Journal of Global Optimization, Springer, vol. 46(1), pages 75-87, January.
    17. Zhi-Long Dong & Fengmin Xu & Yu-Hong Dai, 2020. "Fast algorithms for sparse portfolio selection considering industries and investment styles," Journal of Global Optimization, Springer, vol. 78(4), pages 763-789, December.
    18. Dolgopolik, Maksim V., 2021. "The alternating direction method of multipliers for finding the distance between ellipsoids," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    19. 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.
    20. Bingsheng He & Feng Ma & Xiaoming Yuan, 2020. "Optimally linearizing the alternating direction method of multipliers for convex programming," Computational Optimization and Applications, Springer, vol. 75(2), pages 361-388, March.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:77:y:2020:i:3:d:10.1007_s10589-020-00221-y. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.