IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v81y2022i2d10.1007_s10589-021-00338-8.html
   My bibliography  Save this article

An inexact accelerated stochastic ADMM for separable convex optimization

Author

Listed:
  • Jianchao Bai

    (Northwestern Polytechnical University)

  • William W. Hager

    (University of Florida)

  • Hongchao Zhang

    (Louisiana State University)

Abstract

An inexact accelerated stochastic Alternating Direction Method of Multipliers (AS-ADMM) scheme is developed for solving structured separable convex optimization problems with linear constraints. The objective function is the sum of a possibly nonsmooth convex function and a smooth function which is an average of many component convex functions. Problems having this structure often arise in machine learning and data mining applications. AS-ADMM combines the ideas of both ADMM and the stochastic gradient methods using variance reduction techniques. One of the ADMM subproblems employs a linearization technique while a similar linearization could be introduced for the other subproblem. For a specified choice of the algorithm parameters, it is shown that the objective error and the constraint violation are $$\mathcal {O}(1/k)$$ O ( 1 / k ) relative to the number of outer iterations k. Under a strong convexity assumption, the expected iterate error converges to zero linearly. A linearized variant of AS-ADMM and incremental sampling strategies are also discussed. Numerical experiments with both stochastic and deterministic ADMM algorithms show that AS-ADMM can be particularly effective for structured optimization arising in big data applications.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:coopap:v:81:y:2022:i:2:d:10.1007_s10589-021-00338-8
    DOI: 10.1007/s10589-021-00338-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-021-00338-8
    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/s10589-021-00338-8?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. Min Tao, 2020. "Convergence study of indefinite proximal ADMM with a relaxation factor," Computational Optimization and Applications, Springer, vol. 77(1), pages 91-123, September.
    2. M. H. Xu & T. Wu, 2011. "A Class of Linearized Proximal Alternating Direction Methods," Journal of Optimization Theory and Applications, Springer, vol. 151(2), pages 321-337, November.
    3. 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.
    4. 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.
    5. Jianchao Bai & Jicheng Li & Fengmin Xu & Hongchao Zhang, 2018. "Generalized symmetric ADMM for separable convex optimization," Computational Optimization and Applications, Springer, vol. 70(1), pages 129-170, May.
    6. 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.
    7. 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.
    8. 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.
    9. 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.
    10. Guoyong Gu & Bingsheng He & Junfeng Yang, 2014. "Inexact Alternating-Direction-Based Contraction Methods for Separable Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 163(1), pages 105-129, October.
    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. Albert S. Berahas & Jiahao Shi & Zihong Yi & Baoyu Zhou, 2023. "Accelerating stochastic sequential quadratic programming for equality constrained optimization using predictive variance reduction," Computational Optimization and Applications, Springer, vol. 86(1), pages 79-116, September.

    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, 2020. "Convergence rates for an inexact ADMM applied to separable convex optimization," Computational Optimization and Applications, Springer, vol. 77(3), pages 729-754, December.
    2. 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.
    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. 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.
    5. 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.
    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. Jiao-fen Li & Wen Li & Ru Huang, 2016. "An efficient method for solving a matrix least squares problem over a matrix inequality constraint," Computational Optimization and Applications, Springer, vol. 63(2), pages 393-423, March.
    8. Le Thi Khanh Hien & Duy Nhat Phan & Nicolas Gillis, 2022. "Inertial alternating direction method of multipliers for non-convex non-smooth optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 247-285, September.
    9. 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.
    10. Dolgopolik, Maksim V., 2021. "The alternating direction method of multipliers for finding the distance between ellipsoids," Applied Mathematics and Computation, Elsevier, vol. 409(C).
    11. Tonbari, Mohamed El & Ahmed, Shabbir, 2023. "Consensus-based Dantzig-Wolfe decomposition," European Journal of Operational Research, Elsevier, vol. 307(3), pages 1441-1456.
    12. 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.
    13. Zhongming Wu & Min Li, 2019. "General inertial proximal gradient method for a class of nonconvex nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 129-158, May.
    14. 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.
    15. 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.
    16. Boutaayamou, Idriss & Hadri, Aissam & Laghrib, Amine, 2023. "An optimal bilevel optimization model for the generalized total variation and anisotropic tensor parameters selection," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    17. 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.
    18. Jiaxin Xie, 2018. "On inexact ADMMs with relative error criteria," Computational Optimization and Applications, Springer, vol. 71(3), pages 743-765, December.
    19. 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.
    20. Feng Ma, 2019. "On relaxation of some customized proximal point algorithms for convex minimization: from variational inequality perspective," Computational Optimization and Applications, Springer, vol. 73(3), pages 871-901, July.

    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:81:y:2022:i:2:d:10.1007_s10589-021-00338-8. 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.