IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v147y2010i3d10.1007_s10957-010-9737-7.html
   My bibliography  Save this article

Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization

Author

Listed:
  • S. Sundhar Ram

    (University of Illinois at Urbana-Champaign)

  • A. Nedić

    (University of Illinois at Urbana-Champaign)

  • V. V. Veeravalli

    (University of Illinois at Urbana-Champaign)

Abstract

We consider a distributed multi-agent network system where the goal is to minimize a sum of convex objective functions of the agents subject to a common convex constraint set. Each agent maintains an iterate sequence and communicates the iterates to its neighbors. Then, each agent combines weighted averages of the received iterates with its own iterate, and adjusts the iterate by using subgradient information (known with stochastic errors) of its own function and by projecting onto the constraint set. The goal of this paper is to explore the effects of stochastic subgradient errors on the convergence of the algorithm. We first consider the behavior of the algorithm in mean, and then the convergence with probability 1 and in mean square. We consider general stochastic errors that have uniformly bounded second moments and obtain bounds on the limiting performance of the algorithm in mean for diminishing and non-diminishing stepsizes. When the means of the errors diminish, we prove that there is mean consensus between the agents and mean convergence to the optimum function value for diminishing stepsizes. When the mean errors diminish sufficiently fast, we strengthen the results to consensus and convergence of the iterates to an optimal solution with probability 1 and in mean square.

Suggested Citation

  • S. Sundhar Ram & A. Nedić & V. V. Veeravalli, 2010. "Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 516-545, December.
  • Handle: RePEc:spr:joptap:v:147:y:2010:i:3:d:10.1007_s10957-010-9737-7
    DOI: 10.1007/s10957-010-9737-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-010-9737-7
    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/s10957-010-9737-7?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. M. V. Solodov & S. K. Zavriev, 1998. "Error Stability Properties of Generalized Gradient-Type Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 663-680, September.
    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. Bin Hu & Zhi-Hong Guan & Rui-Quan Liao & Ding-Xue Zhang & Gui-Lin Zheng, 2015. "Consensus-based distributed optimisation of multi-agent networks via a two level subgradient-proximal algorithm," International Journal of Systems Science, Taylor & Francis Journals, vol. 46(7), pages 1307-1318, May.
    2. Woocheol Choi & Doheon Kim & Seok-Bae Yun, 2022. "Convergence Results of a Nested Decentralized Gradient Method for Non-strongly Convex Problems," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 172-204, October.
    3. Jueyou Li & Chuanye Gu & Zhiyou Wu & Changzhi Wu, 2017. "Distributed Optimization Methods for Nonconvex Problems with Inequality Constraints over Time-Varying Networks," Complexity, Hindawi, vol. 2017, pages 1-10, December.
    4. Maude J. Blondin & Matthew Hale, 2021. "A Decentralized Multi-objective Optimization Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 458-485, May.
    5. Junlong Zhu & Ping Xie & Mingchuan Zhang & Ruijuan Zheng & Ling Xing & Qingtao Wu, 2019. "Distributed Stochastic Subgradient Projection Algorithms Based on Weight-Balancing over Time-Varying Directed Graphs," Complexity, Hindawi, vol. 2019, pages 1-16, August.
    6. Haimonti Dutta, 2022. "A Consensus Algorithm for Linear Support Vector Machines," Management Science, INFORMS, vol. 68(5), pages 3703-3725, May.
    7. Wei Ni & Xiaoli Wang, 2022. "A Multi-Scale Method for Distributed Convex Optimization with Constraints," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 379-400, January.
    8. Zhong, Yannan & Xu, Weijun & Li, Hongyi & Zhong, Weiwei, 2024. "Distributed mean reversion online portfolio strategy with stock network," European Journal of Operational Research, Elsevier, vol. 314(3), pages 1143-1158.

    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. Elena Tovbis & Vladimir Krutikov & Predrag Stanimirović & Vladimir Meshechkin & Aleksey Popov & Lev Kazakovtsev, 2023. "A Family of Multi-Step Subgradient Minimization Methods," Mathematics, MDPI, vol. 11(10), pages 1-24, May.
    2. Xiaoliang Wang & Liping Pang & Qi Wu & Mingkun Zhang, 2021. "An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization," Mathematics, MDPI, vol. 9(8), pages 1-27, April.
    3. Jinpeng Ma & Qiongling Li, 2016. "Convergence of price processes under two dynamic double auctions," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 1(1), pages 1-44, December.
    4. Grégory Emiel & Claudia Sagastizábal, 2010. "Incremental-like bundle methods with application to energy planning," Computational Optimization and Applications, Springer, vol. 46(2), pages 305-332, June.
    5. Wenma Jin & Yair Censor & Ming Jiang, 2016. "Bounded perturbation resilience of projected scaled gradient methods," Computational Optimization and Applications, Springer, vol. 63(2), pages 365-392, March.
    6. Regina S. Burachik & Yaohua Hu & Xiaoqi Yang, 2022. "Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces," Journal of Global Optimization, Springer, vol. 83(2), pages 249-271, June.
    7. Larsson, Torbjorn & Patriksson, Michael & Stromberg, Ann-Brith, 2003. "On the convergence of conditional [var epsilon]-subgradient methods for convex programs and convex-concave saddle-point problems," European Journal of Operational Research, Elsevier, vol. 151(3), pages 461-473, December.
    8. Peng Zhang & Gejun Bao, 2018. "An Incremental Subgradient Method on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 711-727, March.
    9. M. V. Solodov, 2003. "On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 151-165, October.
    10. Matthias Rottmann & Kira Maag & Mathis Peyron & Hanno Gottschalk & Nataša Krejić, 2023. "Detection of Iterative Adversarial Attacks via Counter Attack," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 892-929, September.
    11. Xiaojing Xu & Jinpeng Ma & Xiaoping Xie, 2019. "Price Convergence under a Probabilistic Double Auction," Computational Economics, Springer;Society for Computational Economics, vol. 54(3), pages 1113-1155, October.

    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:joptap:v:147:y:2010:i:3:d:10.1007_s10957-010-9737-7. 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.