IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v162y2014i3d10.1007_s10957-013-0491-5.html
   My bibliography  Save this article

The 2-Coordinate Descent Method for Solving Double-Sided Simplex Constrained Minimization Problems

Author

Listed:
  • Amir Beck

    (Technion—Israel Institute of Technology)

Abstract

This paper considers the problem of minimizing a continuously differentiable function with a Lipschitz continuous gradient subject to a single linear equality constraint and additional bound constraints on the decision variables. We introduce and analyze several variants of a 2-coordinate descent method: a block descent method that performs an optimization step with respect to only two variables at each iteration. Based on two new optimality measures, we establish convergence to stationarity points for general nonconvex objective functions. In the convex case, when all the variables are lower bounded but not upper bounded, we show that the sequence of function values converges at a sublinear rate. Several illustrative numerical examples demonstrate the effectiveness of the method.

Suggested Citation

  • Amir Beck, 2014. "The 2-Coordinate Descent Method for Solving Double-Sided Simplex Constrained Minimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 892-919, September.
    • Handle: RePEc:spr:joptap:v:162:y:2014:i:3:d:10.1007_s10957-013-0491-5
    DOI: 10.1007/s10957-013-0491-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-013-0491-5
    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/s10957-013-0491-5?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. C. J. Lin & S. Lucidi & L. Palagi & A. Risi & M. Sciandrone, 2009. "Decomposition Algorithm Model for Singly Linearly-Constrained Problems Subject to Lower and Upper Bounds," Journal of Optimization Theory and Applications, Springer, vol. 141(1), pages 107-126, April.
    2. Paul Tseng & Sangwoon Yun, 2010. "A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training," Computational Optimization and Applications, Springer, vol. 47(2), pages 179-206, October.
    3. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
    4. Cassioli, A. & Di Lorenzo, D. & Sciandrone, M., 2013. "On the convergence of inexact block coordinate descent methods for constrained optimization," European Journal of Operational Research, Elsevier, vol. 231(2), pages 274-281.
    5. Giampaolo Liuzzi & Laura Palagi & Mauro Piacentini, 2010. "On the convergence of a Jacobi-type algorithm for Singly Linearly-Constrained Problems Subject to simple Bounds," DIS Technical Reports 2010-01, Department of Computer, Control and Management Engineering, Universita' degli Studi di Roma "La Sapienza".
    6. NESTEROV, Yurii, 2010. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Discussion Papers CORE 2010002, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    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. A. Ghaffari-Hadigheh & L. Sinjorgo & R. Sotirov, 2024. "On convergence of a q-random coordinate constrained algorithm for non-convex problems," Journal of Global Optimization, Springer, vol. 90(4), pages 843-868, December.
    2. Cristofari, Andrea, 2023. "A decomposition method for lasso problems with zero-sum constraint," European Journal of Operational Research, Elsevier, vol. 306(1), pages 358-369.
    3. Andrea Cristofari, 2019. "An almost cyclic 2-coordinate descent method for singly linearly constrained problems," Computational Optimization and Applications, Springer, vol. 73(2), pages 411-452, June.
    4. Ion Necoara & Yurii Nesterov & François Glineur, 2017. "Random Block Coordinate Descent Methods for Linearly Constrained Optimization over Networks," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 227-254, April.
    5. I. V. Konnov, 2016. "Selective bi-coordinate variations for resource allocation type problems," Computational Optimization and Applications, Springer, vol. 64(3), pages 821-842, July.

    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. Ion Necoara & Andrei Patrascu, 2014. "A random coordinate descent algorithm for optimization problems with composite objective function and linear coupled constraints," Computational Optimization and Applications, Springer, vol. 57(2), pages 307-337, March.
    2. Andrea Manno & Laura Palagi & Simone Sagratella, 2018. "Parallel decomposition methods for linearly constrained problems subject to simple bound with application to the SVMs training," Computational Optimization and Applications, Springer, vol. 71(1), pages 115-145, September.
    3. Kimon Fountoulakis & Rachael Tappenden, 2018. "A flexible coordinate descent method," Computational Optimization and Applications, Springer, vol. 70(2), pages 351-394, June.
    4. Leonardo Galli & Alessandro Galligari & Marco Sciandrone, 2020. "A unified convergence framework for nonmonotone inexact decomposition methods," Computational Optimization and Applications, Springer, vol. 75(1), pages 113-144, January.
    5. Andrea Cristofari, 2019. "An almost cyclic 2-coordinate descent method for singly linearly constrained problems," Computational Optimization and Applications, Springer, vol. 73(2), pages 411-452, June.
    6. 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.
    7. Andrei Patrascu & Ion Necoara, 2015. "Efficient random coordinate descent algorithms for large-scale structured nonconvex optimization," Journal of Global Optimization, Springer, vol. 61(1), pages 19-46, January.
    8. Andrea Manno & Laura Palagi & Simone Sagratella, 2014. "A Class of Convergent Parallel Algorithms for SVMs Training," DIAG Technical Reports 2014-17, Department of Computer, Control and Management Engineering, Universita' degli Studi di Roma "La Sapienza".
    9. Tommaso Colombo & Simone Sagratella, 2020. "Distributed algorithms for convex problems with linear coupling constraints," Journal of Global Optimization, Springer, vol. 77(1), pages 53-73, May.
    10. Wei Peng & Hui Zhang & Xiaoya Zhang, 2019. "Nonconvex Proximal Incremental Aggregated Gradient Method with Linear Convergence," Journal of Optimization Theory and Applications, Springer, vol. 183(1), pages 230-245, October.
    11. Cassioli, A. & Di Lorenzo, D. & Sciandrone, M., 2013. "On the convergence of inexact block coordinate descent methods for constrained optimization," European Journal of Operational Research, Elsevier, vol. 231(2), pages 274-281.
    12. Rachael Tappenden & Peter Richtárik & Jacek Gondzio, 2016. "Inexact Coordinate Descent: Complexity and Preconditioning," Journal of Optimization Theory and Applications, Springer, vol. 170(1), pages 144-176, July.
    13. Min Tao & Jiang-Ning Li, 2023. "Error Bound and Isocost Imply Linear Convergence of DCA-Based Algorithms to D-Stationarity," Journal of Optimization Theory and Applications, Springer, vol. 197(1), pages 205-232, April.
    14. 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.
    15. Weizhe Gu & Wei-Po Chen & Chun-Hsu Ko & Yuh-Jye Lee & Jein-Shan Chen, 2018. "Two smooth support vector machines for $$\varepsilon $$ ε -insensitive regression," Computational Optimization and Applications, Springer, vol. 70(1), pages 171-199, May.
    16. Dewei Zhang & Yin Liu & Sam Davanloo Tajbakhsh, 2022. "A First-Order Optimization Algorithm for Statistical Learning with Hierarchical Sparsity Structure," INFORMS Journal on Computing, INFORMS, vol. 34(2), pages 1126-1140, March.
    17. Liu, Yulan & Bi, Shujun, 2019. "Error bounds for non-polyhedral convex optimization and applications to linear convergence of FDM and PGM," Applied Mathematics and Computation, Elsevier, vol. 358(C), pages 418-435.
    18. Paul Tseng & Sangwoon Yun, 2010. "A coordinate gradient descent method for linearly constrained smooth optimization and support vector machines training," Computational Optimization and Applications, Springer, vol. 47(2), pages 179-206, October.
    19. Sjur Didrik Flåm, 2019. "Blocks of coordinates, stochastic programming, and markets," Computational Management Science, Springer, vol. 16(1), pages 3-16, February.
    20. Pei Wang & Shunjie Chen & Sijia Yang, 2022. "Recent Advances on Penalized Regression Models for Biological Data," Mathematics, MDPI, vol. 10(19), pages 1-24, October.

    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:162:y:2014:i:3:d:10.1007_s10957-013-0491-5. 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.