IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v54y2013i2p417-440.html
   My bibliography  Save this article

A cyclic projected gradient method

Author

Listed:
  • Simon Setzer
  • Gabriele Steidl
  • Jan Morgenthaler

Abstract

In recent years, convex optimization methods were successfully applied for various image processing tasks and a large number of first-order methods were designed to minimize the corresponding functionals. Interestingly, it was shown recently in Grewenig et al. ( 2010 ) that the simple idea of so-called “superstep cycles” leads to very efficient schemes for time-dependent (parabolic) image enhancement problems as well as for steady state (elliptic) image compression tasks. The “superstep cycles” approach is similar to the nonstationary (cyclic) Richardson method which has been around for over sixty years. In this paper, we investigate the incorporation of superstep cycles into the projected gradient method. We show for two problems in compressive sensing and image processing, namely the LASSO approach and the Rudin-Osher-Fatemi model that the resulting simple cyclic projected gradient algorithm can numerically compare with various state-of-the-art first-order algorithms. However, due to the nonlinear projection within the algorithm convergence proofs even under restrictive assumptions on the linear operators appear to be hard. We demonstrate the difficulties by studying the simplest case of a two-cycle algorithm in ℝ 2 with projections onto the Euclidean ball. Copyright Springer Science+Business Media New York 2013

Suggested Citation

  • Simon Setzer & Gabriele Steidl & Jan Morgenthaler, 2013. "A cyclic projected gradient method," Computational Optimization and Applications, Springer, vol. 54(2), pages 417-440, March.
  • Handle: RePEc:spr:coopap:v:54:y:2013:i:2:p:417-440
    DOI: 10.1007/s10589-012-9525-4
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s10589-012-9525-4
    Download Restriction: Access to full text is restricted to subscribers.

    File URL: https://libkey.io/10.1007/s10589-012-9525-4?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., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    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. Masaru Ito, 2016. "New results on subgradient methods for strongly convex optimization problems with a unified analysis," Computational Optimization and Applications, Springer, vol. 65(1), pages 127-172, September.
    2. TAYLOR, Adrien B. & HENDRICKX, Julien M. & François GLINEUR, 2016. "Exact worst-case performance of first-order methods for composite convex optimization," LIDAM Discussion Papers CORE 2016052, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. Dimitris Bertsimas & Nishanth Mundru, 2021. "Sparse Convex Regression," INFORMS Journal on Computing, INFORMS, vol. 33(1), pages 262-279, January.
    4. Alexandre Belloni & Victor Chernozhukov & Lie Wang, 2013. "Pivotal estimation via square-root lasso in nonparametric regression," CeMMAP working papers CWP62/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    5. DEVOLDER, Olivier & GLINEUR, François & NESTEROV, Yurii, 2013. "First-order methods with inexact oracle: the strongly convex case," LIDAM Discussion Papers CORE 2013016, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    6. Chao, Shih-Kang & Härdle, Wolfgang K. & Yuan, Ming, 2021. "Factorisable Multitask Quantile Regression," Econometric Theory, Cambridge University Press, vol. 37(4), pages 794-816, August.
    7. David Degras, 2021. "Sparse group fused lasso for model segmentation: a hybrid approach," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 15(3), pages 625-671, September.
    8. Yunmei Chen & Xiaojing Ye & Wei Zhang, 2020. "Acceleration techniques for level bundle methods in weakly smooth convex constrained optimization," Computational Optimization and Applications, Springer, vol. 77(2), pages 411-432, November.
    9. Silvia Villa & Lorenzo Rosasco & Sofia Mosci & Alessandro Verri, 2014. "Proximal methods for the latent group lasso penalty," Computational Optimization and Applications, Springer, vol. 58(2), pages 381-407, June.
    10. Wenjie Huang & Xun Zhang, 2021. "Randomized Smoothing Variance Reduction Method for Large-Scale Non-smooth Convex Optimization," SN Operations Research Forum, Springer, vol. 2(2), pages 1-28, June.
    11. Le Thi Khanh Hien & Cuong V. Nguyen & Huan Xu & Canyi Lu & Jiashi Feng, 2019. "Accelerated Randomized Mirror Descent Algorithms for Composite Non-strongly Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 181(2), pages 541-566, May.
    12. DEVOLDER, Olivier, 2011. "Stochastic first order methods in smooth convex optimization," LIDAM Discussion Papers CORE 2011070, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    13. Boriss Siliverstovs, 2017. "Short-term forecasting with mixed-frequency data: a MIDASSO approach," Applied Economics, Taylor & Francis Journals, vol. 49(13), pages 1326-1343, March.
    14. 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.
    15. David Müller & Vladimir Shikhman, 2022. "Network manipulation algorithm based on inexact alternating minimization," Computational Management Science, Springer, vol. 19(4), pages 627-664, October.
    16. Gondzio, Jacek, 2012. "Interior point methods 25 years later," European Journal of Operational Research, Elsevier, vol. 218(3), pages 587-601.
    17. A. Chambolle & Ch. Dossal, 2015. "On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 968-982, September.
    18. Nima Rabiei & Jose Muñoz, 2015. "AAR-based decomposition algorithm for non-linear convex optimisation," Computational Optimization and Applications, Springer, vol. 62(3), pages 761-786, December.
    19. ARAVENA, Ignacio & PAPAVASILIOU, Anthony, 2016. "An Asynchronous Distributed Algorithm for solving Stochastic Unit Commitment," LIDAM Discussion Papers CORE 2016038, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    20. Pi, J. & Wang, Honggang & Pardalos, Panos M., 2021. "A dual reformulation and solution framework for regularized convex clustering problems," European Journal of Operational Research, Elsevier, vol. 290(3), pages 844-856.

    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:54:y:2013:i:2:p:417-440. 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.