IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v62y2015i2p373-404.html
   My bibliography  Save this article

Inexact accelerated augmented Lagrangian methods

Author

Listed:
  • Myeongmin Kang
  • Myungjoo Kang
  • Miyoun Jung

Abstract

The augmented Lagrangian method (ALM) is a popular method for solving linearly constrained convex minimization problems, and it has been used in many applications such as compressive sensing or image processing. Recently, accelerated versions of the augmented Lagrangian method (AALM) have been developed, and they assume that the subproblem can be exactly solved. However, the subproblem of the augmented Lagrangian method in general does not have a closed-form solution. In this paper, we introduce an inexact version of an accelerated augmented Lagrangian method (I-AALM), with an implementable inexact stopping condition for the subproblem. It is also proved that the convergence rate of our method remains the same as the accelerated ALM, which is $${{\mathcal {O}}}(\frac{1}{k^2})$$ O ( 1 k 2 ) with an iteration number $$k$$ k . In a similar manner, we propose an inexact accelerated alternating direction method of multiplier (I-AADMM), which is an inexact version of an accelerated ADMM. Numerical applications to compressive sensing or image inpainting are also presented to validate the effectiveness of the proposed iterative algorithms. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • Myeongmin Kang & Myungjoo Kang & Miyoun Jung, 2015. "Inexact accelerated augmented Lagrangian methods," Computational Optimization and Applications, Springer, vol. 62(2), pages 373-404, November.
  • Handle: RePEc:spr:coopap:v:62:y:2015:i:2:p:373-404
    DOI: 10.1007/s10589-015-9742-8
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-015-9742-8
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s10589-015-9742-8?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. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Bingsheng He & Xiaoming Yuan, 2012. "An Accelerated Inexact Proximal Point Algorithm for Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 154(2), pages 536-548, August.
    3. 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. Hedy Attouch & Zaki Chbani & Jalal Fadili & Hassan Riahi, 2022. "Fast Convergence of Dynamical ADMM via Time Scaling of Damped Inertial Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 704-736, June.

    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. Julian Rasch & Antonin Chambolle, 2020. "Inexact first-order primal–dual algorithms," Computational Optimization and Applications, Springer, vol. 76(2), pages 381-430, June.
    2. Guoqiang Wang & Bo Yu, 2019. "PAL-Hom method for QP and an application to LP," Computational Optimization and Applications, Springer, vol. 73(1), pages 311-352, May.
    3. 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.
    4. 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.
    5. Umberto Amato & Anestis Antoniadis & Italia De Feis & Irene Gijbels, 2021. "Penalised robust estimators for sparse and high-dimensional linear models," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 30(1), pages 1-48, March.
    6. Kohli, Priya & Garcia, Tanya P. & Pourahmadi, Mohsen, 2016. "Modeling the Cholesky factors of covariance matrices of multivariate longitudinal data," Journal of Multivariate Analysis, Elsevier, vol. 145(C), pages 87-100.
    7. 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.
    8. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2011. "First-order methods of smooth convex optimization with inexact oracle," LIDAM Discussion Papers CORE 2011002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    9. Jiahe Lin & George Michailidis, 2019. "Approximate Factor Models with Strongly Correlated Idiosyncratic Errors," Papers 1912.04123, arXiv.org.
    10. 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.
    11. 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.
    12. 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.
    13. Mingqiang Li & Congying Han & Ruxin Wang & Tiande Guo, 2017. "Shrinking gradient descent algorithms for total variation regularized image denoising," Computational Optimization and Applications, Springer, vol. 68(3), pages 643-660, December.
    14. 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.
    15. 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.
    16. Hansen, Christian & Liao, Yuan, 2019. "The Factor-Lasso And K-Step Bootstrap Approach For Inference In High-Dimensional Economic Applications," Econometric Theory, Cambridge University Press, vol. 35(3), pages 465-509, June.
    17. 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.
    18. 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.
    19. Jueyou Li & Zhiyou Wu & Changzhi Wu & Qiang Long & Xiangyu Wang, 2016. "An Inexact Dual Fast Gradient-Projection Method for Separable Convex Optimization with Linear Coupled Constraints," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 153-171, January.
    20. 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.

    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:62:y:2015:i:2:p:373-404. 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.