IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v68y2017i2d10.1007_s10589-017-9911-z.html
   My bibliography  Save this article

Approximate ADMM algorithms derived from Lagrangian splitting

Author

Listed:
  • Jonathan Eckstein

    (Rutgers University)

  • Wang Yao

    (Rutgers University)

Abstract

This paper presents two new approximate versions of the alternating direction method of multipliers (ADMM) derived by modifying of the original “Lagrangian splitting” convergence analysis of Fortin and Glowinski. They require neither strong convexity of the objective function nor any restrictions on the coupling matrix. The first method uses an absolutely summable error criterion and resembles methods that may readily be derived from earlier work on the relationship between the ADMM and the proximal point method, but without any need for restrictive assumptions to make it practically implementable. It permits both subproblems to be solved inexactly. The second method uses a relative error criterion and the same kind of auxiliary iterate sequence that has recently been proposed to enable relative-error approximate implementation of non-decomposition augmented Lagrangian algorithms. It also allows both subproblems to be solved inexactly, although ruling out “jamming” behavior requires a somewhat complicated implementation. The convergence analyses of the two methods share extensive underlying elements.

Suggested Citation

  • Jonathan Eckstein & Wang Yao, 2017. "Approximate ADMM algorithms derived from Lagrangian splitting," Computational Optimization and Applications, Springer, vol. 68(2), pages 363-405, November.
    • Handle: RePEc:spr:coopap:v:68:y:2017:i:2:d:10.1007_s10589-017-9911-z
    DOI: 10.1007/s10589-017-9911-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-017-9911-z
    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/s10589-017-9911-z?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. Dettling, Marcel & Bühlmann, Peter, 2004. "Finding predictive gene groups from microarray data," Journal of Multivariate Analysis, Elsevier, vol. 90(1), pages 106-131, July.
    2. 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.
    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. 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. 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.
    3. Vando A. Adona & Max L. N. Gonçalves & Jefferson G. Melo, 2019. "A Partially Inexact Proximal Alternating Direction Method of Multipliers and Its Iteration-Complexity Analysis," Journal of Optimization Theory and Applications, Springer, vol. 182(2), pages 640-666, August.
    4. Yunier Bello-Cruz & Max L. N. Gonçalves & Nathan Krislock, 2023. "On FISTA with a relative error rule," Computational Optimization and Applications, Springer, vol. 84(2), pages 295-318, March.
    5. Jiaxin Xie, 2018. "On inexact ADMMs with relative error criteria," Computational Optimization and Applications, Springer, vol. 71(3), pages 743-765, December.
    6. M. Marques Alves & Jonathan Eckstein & Marina Geremia & Jefferson G. Melo, 2020. "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," Computational Optimization and Applications, Springer, vol. 75(2), pages 389-422, March.
    7. V. A. Adona & M. L. N. Gonçalves & J. G. Melo, 2020. "An inexact proximal generalized alternating direction method of multipliers," Computational Optimization and Applications, Springer, vol. 76(3), pages 621-647, July.
    8. 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.

    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. Mostafa Rezaei & Ivor Cribben & Michele Samorani, 2021. "A clustering-based feature selection method for automatically generated relational attributes," Annals of Operations Research, Springer, vol. 303(1), pages 233-263, August.
    2. 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.
    3. Zambom, Adriano Zanin & Akritas, Michael G., 2015. "Nonparametric significance testing and group variable selection," Journal of Multivariate Analysis, Elsevier, vol. 133(C), pages 51-60.
    4. 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.
    5. Jessie J Hsu & Dianne M Finkelstein & David A Schoenfeld, 2015. "Outcome-Driven Cluster Analysis with Application to Microarray Data," PLOS ONE, Public Library of Science, vol. 10(11), pages 1-15, November.
    6. Renato D. C. Monteiro & Chee-Khian Sim, 2018. "Complexity of the relaxed Peaceman–Rachford splitting method for the sum of two maximal strongly monotone operators," Computational Optimization and Applications, Springer, vol. 70(3), pages 763-790, July.
    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. Jonathan Eckstein & Paulo Silva, 2010. "Proximal methods for nonlinear programming: double regularization and inexact subproblems," Computational Optimization and Applications, Springer, vol. 46(2), pages 279-304, June.
    9. Cui, Qiurong & Xu, Yuqing & Zhang, Zhengjun & Chan, Vincent, 2021. "Max-linear regression models with regularization," Journal of Econometrics, Elsevier, vol. 222(1), pages 579-600.
    10. Maicon Marques Alves & Samara Costa Lima, 2017. "An Inexact Spingarn’s Partial Inverse Method with Applications to Operator Splitting and Composite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 175(3), pages 818-847, December.
    11. Garcia-Magariños Manuel & Antoniadis Anestis & Cao Ricardo & González-Manteiga Wenceslao, 2010. "Lasso Logistic Regression, GSoft and the Cyclic Coordinate Descent Algorithm: Application to Gene Expression Data," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 9(1), pages 1-30, August.
    12. Ceng, Lu-Chuan & Yao, Jen-Chih, 2007. "Approximate proximal methods in vector optimization," European Journal of Operational Research, Elsevier, vol. 183(1), pages 1-19, November.
    13. Lu-Chuan Ceng & Nicolas Hadjisavvas & Ngai-Ching Wong, 2010. "Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems," Journal of Global Optimization, Springer, vol. 46(4), pages 635-646, April.
    14. Howard D. Bondell & Brian J. Reich, 2008. "Simultaneous Regression Shrinkage, Variable Selection, and Supervised Clustering of Predictors with OSCAR," Biometrics, The International Biometric Society, vol. 64(1), pages 115-123, March.
    15. M. Marques Alves & Jonathan Eckstein & Marina Geremia & Jefferson G. Melo, 2020. "Relative-error inertial-relaxed inexact versions of Douglas-Rachford and ADMM splitting algorithms," Computational Optimization and Applications, Springer, vol. 75(2), pages 389-422, March.
    16. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    17. Yao, Xingzhi & Izzeldin, Marwan & Li, Zhenxiong, 2019. "A novel cluster HAR-type model for forecasting realized volatility," International Journal of Forecasting, Elsevier, vol. 35(4), pages 1318-1331.
    18. Souza, Sissy da S. & Oliveira, P.R. & da Cruz Neto, J.X. & Soubeyran, A., 2010. "A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 201(2), pages 365-376, March.
    19. Nils Langenberg, 2012. "An Interior Proximal Method for a Class of Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 902-922, December.
    20. Jiaxin Xie, 2018. "On inexact ADMMs with relative error criteria," Computational Optimization and Applications, Springer, vol. 71(3), pages 743-765, December.

    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:68:y:2017:i:2:d:10.1007_s10589-017-9911-z. 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.