IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v70y2018i3d10.1007_s10589-018-9992-3.html
   My bibliography  Save this article

The generalized proximal point algorithm with step size 2 is not necessarily convergent

Author

Listed:
  • Min Tao

    (Nanjing University)

  • Xiaoming Yuan

    (The University of Hong Kong)

Abstract

The proximal point algorithm (PPA) is a fundamental method in optimization and it has been well studied in the literature. Recently a generalized version of the PPA with a step size in (0, 2) has been proposed. Inheriting all important theoretical properties of the original PPA, the generalized PPA has some numerical advantages that have been well verified in the literature by various applications. A common sense is that larger step sizes are preferred whenever the convergence can be theoretically ensured; thus it is interesting to know whether or not the step size of the generalized PPA can be as large as 2. We give a negative answer to this question. Some counterexamples are constructed to illustrate the divergence of the generalized PPA with step size 2 in both generic and specific settings, including the generalized versions of the very popular augmented Lagrangian method and the alternating direction method of multipliers. A by-product of our analysis is the failure of convergence of the Peaceman–Rachford splitting method and a generalized version of the forward–backward splitting method with step size 1.5.

Suggested Citation

  • Min Tao & Xiaoming Yuan, 2018. "The generalized proximal point algorithm with step size 2 is not necessarily convergent," Computational Optimization and Applications, Springer, vol. 70(3), pages 827-839, July.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:3:d:10.1007_s10589-018-9992-3
    DOI: 10.1007/s10589-018-9992-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-018-9992-3
    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/s10589-018-9992-3?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. R. T. Rockafellar, 1976. "Augmented Lagrangians and Applications of the Proximal Point Algorithm in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 97-116, May.
    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. Min Tao & Xiaoming Yuan, 2018. "On Glowinski’s Open Question on the Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 163-196, October.

    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. Mauricio Romero Sicre, 2020. "On the complexity of a hybrid proximal extragradient projective method for solving monotone inclusion problems," Computational Optimization and Applications, Springer, vol. 76(3), pages 991-1019, July.
    2. Jean-Pierre Crouzeix & Abdelhak Hassouni & Eladio Ocaña, 2023. "A Short Note on the Twice Differentiability of the Marginal Function of a Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 857-867, August.
    3. Liang Chen & Anping Liao, 2020. "On the Convergence Properties of a Second-Order Augmented Lagrangian Method for Nonlinear Programming Problems with Inequality Constraints," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 248-265, October.
    4. Stefano Cipolla & Jacek Gondzio, 2023. "Proximal Stabilized Interior Point Methods and Low-Frequency-Update Preconditioning Techniques," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1061-1103, June.
    5. Bingsheng He & Li-Zhi Liao & Xiang Wang, 2012. "Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods," Computational Optimization and Applications, Springer, vol. 51(2), pages 649-679, March.
    6. Marwan A. Kutbi & Abdul Latif & Xiaolong Qin, 2019. "Convergence of Two Splitting Projection Algorithms in Hilbert Spaces," Mathematics, MDPI, vol. 7(10), pages 1-13, October.
    7. Darinka Dentcheva & Gabriela Martinez & Eli Wolfhagen, 2016. "Augmented Lagrangian Methods for Solving Optimization Problems with Stochastic-Order Constraints," Operations Research, INFORMS, vol. 64(6), pages 1451-1465, December.
    8. Gui-Hua Lin & Zhen-Ping Yang & Hai-An Yin & Jin Zhang, 2023. "A dual-based stochastic inexact algorithm for a class of stochastic nonsmooth convex composite problems," Computational Optimization and Applications, Springer, vol. 86(2), pages 669-710, November.
    9. Xiaoming Yuan, 2011. "An improved proximal alternating direction method for monotone variational inequalities with separable structure," Computational Optimization and Applications, Springer, vol. 49(1), pages 17-29, May.
    10. Zhu, Daoli & Marcotte, Patrice, 1995. "Coupling the auxiliary problem principle with descent methods of pseudoconvex programming," European Journal of Operational Research, Elsevier, vol. 83(3), pages 670-685, June.
    11. Guo, Zhaomiao & Fan, Yueyue, 2017. "A Stochastic Multi-Agent Optimization Model for Energy Infrastructure Planning Under Uncertainty and Competition," Institute of Transportation Studies, Working Paper Series qt89s5s8hn, Institute of Transportation Studies, UC Davis.
    12. Yong-Jin Liu & Jing Yu, 2023. "A semismooth Newton based dual proximal point algorithm for maximum eigenvalue problem," Computational Optimization and Applications, Springer, vol. 85(2), pages 547-582, June.
    13. Julian Rasch & Antonin Chambolle, 2020. "Inexact first-order primal–dual algorithms," Computational Optimization and Applications, Springer, vol. 76(2), pages 381-430, June.
    14. A. Ruszczynski, 1994. "On Augmented Lagrangian Decomposition Methods For Multistage Stochastic Programs," Working Papers wp94005, International Institute for Applied Systems Analysis.
    15. Liwei Zhang & Yule Zhang & Jia Wu & Xiantao Xiao, 2022. "Solving Stochastic Optimization with Expectation Constraints Efficiently by a Stochastic Augmented Lagrangian-Type Algorithm," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 2989-3006, November.
    16. R. S. Burachik & S. Scheimberg & B. F. Svaiter, 2001. "Robustness of the Hybrid Extragradient Proximal-Point Algorithm," Journal of Optimization Theory and Applications, Springer, vol. 111(1), pages 117-136, October.
    17. A. F. Izmailov & M. V. Solodov, 2022. "Perturbed Augmented Lagrangian Method Framework with Applications to Proximal and Smoothed Variants," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 491-522, June.
    18. Pinheiro, Ricardo B.N.M. & Lage, Guilherme G. & da Costa, Geraldo R.M., 2019. "A primal-dual integrated nonlinear rescaling approach applied to the optimal reactive dispatch problem," European Journal of Operational Research, Elsevier, vol. 276(3), pages 1137-1153.
    19. M. Kyono & M. Fukushima, 2000. "Nonlinear Proximal Decomposition Method for Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 106(2), pages 357-372, August.
    20. Ya-Feng Liu & Xin Liu & Shiqian Ma, 2019. "On the Nonergodic Convergence Rate of an Inexact Augmented Lagrangian Framework for Composite Convex Programming," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 632-650, May.

    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:70:y:2018:i:3:d:10.1007_s10589-018-9992-3. 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.