IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v182y2019i1d10.1007_s10957-018-01454-y.html
   My bibliography  Save this article

The Proximal Alternating Minimization Algorithm for Two-Block Separable Convex Optimization Problems with Linear Constraints

Author

Listed:
  • Sandy Bitterlich

    (Chemnitz University of Technology)

  • Radu Ioan Boţ

    (University of Vienna)

  • Ernö Robert Csetnek

    (University of Vienna
    University of Göttingen)

  • Gert Wanka

    (Chemnitz University of Technology)

Abstract

The Alternating Minimization Algorithm has been proposed by Paul Tseng to solve convex programming problems with two-block separable linear constraints and objectives, whereby (at least) one of the components of the latter is assumed to be strongly convex. The fact that one of the subproblems to be solved within the iteration process of this method does not usually correspond to the calculation of a proximal operator through a closed formula affects the implementability of the algorithm. In this paper, we allow in each block of the objective a further smooth convex function and propose a proximal version of the algorithm, which is achieved by equipping the algorithm with proximal terms induced by variable metrics. For suitable choices of the latter, the solving of the two subproblems in the iterative scheme can be reduced to the computation of proximal operators. We investigate the convergence of the proposed algorithm in a real Hilbert space setting and illustrate its numerical performances on two applications in image processing and machine learning.

Suggested Citation

  • Sandy Bitterlich & Radu Ioan Boţ & Ernö Robert Csetnek & Gert Wanka, 2019. "The Proximal Alternating Minimization Algorithm for Two-Block Separable Convex Optimization Problems with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 110-132, July.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-018-01454-y
    DOI: 10.1007/s10957-018-01454-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-018-01454-y
    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/s10957-018-01454-y?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. Kristian Bredies & Hongpeng Sun, 2017. "A Proximal Point Analysis of the Preconditioned Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 878-907, June.
    2. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    3. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2016. "A block coordinate variable metric forward–backward algorithm," Journal of Global Optimization, Springer, vol. 66(3), pages 457-485, November.
    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. Silvia Bonettini & Peter Ochs & Marco Prato & Simone Rebegoldi, 2023. "An abstract convergence framework with application to inertial inexact forward–backward methods," Computational Optimization and Applications, Springer, vol. 84(2), pages 319-362, March.
    2. S. Bonettini & M. Prato & S. Rebegoldi, 2018. "A block coordinate variable metric linesearch based proximal gradient method," Computational Optimization and Applications, Springer, vol. 71(1), pages 5-52, September.
    3. Pooja Gupta & Angshul Majumdar & Emilie Chouzenoux & Giovanni Chierchia, 2020. "SuperDeConFuse: A Supervised Deep Convolutional Transform based Fusion Framework for Financial Trading Systems," Papers 2011.04364, arXiv.org.
    4. Guillaume Sagnol & Edouard Pauwels, 2019. "An unexpected connection between Bayes A-optimal designs and the group lasso," Statistical Papers, Springer, vol. 60(2), pages 565-584, April.
    5. Ernest K. Ryu & Yanli Liu & Wotao Yin, 2019. "Douglas–Rachford splitting and ADMM for pathological convex optimization," Computational Optimization and Applications, Springer, vol. 74(3), pages 747-778, December.
    6. Junhong Lin & Lorenzo Rosasco & Silvia Villa & Ding-Xuan Zhou, 2018. "Modified Fejér sequences and applications," Computational Optimization and Applications, Springer, vol. 71(1), pages 95-113, September.
    7. Puya Latafat & Panagiotis Patrinos, 2017. "Asymmetric forward–backward–adjoint splitting for solving monotone inclusions involving three operators," Computational Optimization and Applications, Springer, vol. 68(1), pages 57-93, September.
    8. Le Thi Khanh Hien & Duy Nhat Phan & Nicolas Gillis, 2022. "Inertial alternating direction method of multipliers for non-convex non-smooth optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 247-285, September.
    9. Sedi Bartz & Rubén Campoy & Hung M. Phan, 2022. "An Adaptive Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1019-1055, December.
    10. Hedy Attouch & Alexandre Cabot & Zaki Chbani & Hassan Riahi, 2018. "Inertial Forward–Backward Algorithms with Perturbations: Application to Tikhonov Regularization," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 1-36, October.
    11. 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).
    12. Suthep Suantai & Kunrada Kankam & Prasit Cholamjiak, 2020. "A Novel Forward-Backward Algorithm for Solving Convex Minimization Problem in Hilbert Spaces," Mathematics, MDPI, vol. 8(1), pages 1-13, January.
    13. Hongpeng Sun, 2019. "Analysis of Fully Preconditioned Alternating Direction Method of Multipliers with Relaxation in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 199-229, October.
    14. Wang, Yugang & Huang, Ting-Zhu & Zhao, Xi-Le & Deng, Liang-Jian & Ji, Teng-Yu, 2020. "A convex single image dehazing model via sparse dark channel prior," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    15. Julian Rasch & Antonin Chambolle, 2020. "Inexact first-order primal–dual algorithms," Computational Optimization and Applications, Springer, vol. 76(2), pages 381-430, June.
    16. Peter Ochs, 2018. "Local Convergence of the Heavy-Ball Method and iPiano for Non-convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 153-180, April.
    17. Sun, Shilin & Wang, Tianyang & Yang, Hongxing & Chu, Fulei, 2022. "Damage identification of wind turbine blades using an adaptive method for compressive beamforming based on the generalized minimax-concave penalty function," Renewable Energy, Elsevier, vol. 181(C), pages 59-70.
    18. 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.
    19. Anda Tang & Pei Quan & Lingfeng Niu & Yong Shi, 2022. "A Survey for Sparse Regularization Based Compression Methods," Annals of Data Science, Springer, vol. 9(4), pages 695-722, August.
    20. Christian Grussler & Pontus Giselsson, 2022. "Efficient Proximal Mapping Computation for Low-Rank Inducing Norms," Journal of Optimization Theory and Applications, Springer, vol. 192(1), pages 168-194, January.

    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:182:y:2019:i:1:d:10.1007_s10957-018-01454-y. 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.