IDEAS home Printed from https://ideas.repec.org/p/cor/louvco/2016027.html
   My bibliography  Save this paper

On the Worst-case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions

Author

Listed:
  • DE KLERK, Etienne
  • GLINEUR, François

    (Université catholique de Louvain, CORE, Belgium)

  • TAYLOR, Adrien B.

Abstract

We consider the gradient (or steepest) descent method with exact line search applied to a strongly convex function with Lipschitz continuous gradient. We establish the exact worst-case rate of convergence of this scheme, and show that this worst-case behavior is exhibited by a certain convex quadratic function. We also extend the result to a noisy variant of gradient descent method, where exact line-search is performed in a search direction that differs from negative gradient by at most a prescribed relative tolerance. The proof is computer-assisted, and relies on the resolution of semidefinite programming performance estimation problems as introduced in the paper [Y. Drori and M. Teboulle. Performance of first-order methods for smooth convex minimization: a novel approach. Mathematical Programming, 145(1-2):451-482, 2014].

Suggested Citation

  • DE KLERK, Etienne & GLINEUR, François & TAYLOR, Adrien B., 2016. "On the Worst-case Complexity of the Gradient Method with Exact Line Search for Smooth Strongly Convex Functions," LIDAM Discussion Papers CORE 2016027, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    • Handle: RePEc:cor:louvco:2016027
    as

    Download full text from publisher

    File URL: https://sites.uclouvain.be/core/publications/coredp/coredp2016.html
    Download Restriction: no
    ---><---

    Other versions of this item:

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Roland Hildebrand, 2021. "Optimal step length for the Newton method: case of self-concordant functions," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(2), pages 253-279, October.
    2. André Uschmajew & Bart Vandereycken, 2022. "A Note on the Optimal Convergence Rate of Descent Methods with Fixed Step Sizes for Smooth Strongly Convex Functions," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 364-373, July.
    3. Sandra S. Y. Tan & Antonios Varvitsiotis & Vincent Y. F. Tan, 2021. "Analysis of Optimization Algorithms via Sum-of-Squares," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 56-81, July.
    4. Adrien B. Taylor & Julien M. Hendrickx & François Glineur, 2018. "Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 455-476, August.
    5. Abbaszadehpeivasti, Hadi & de Klerk, Etienne & Zamani, Moslem, 2022. "The exact worst-case convergence rate of the gradient method with fixed step lengths for L-smooth functions," Other publications TiSEM 061688c6-f97c-4024-bb5b-1, Tilburg University, School of Economics and Management.

    More about this item

    Keywords

    gradient method; steepest descent; semidefinite programming; performance estimation problem;
    All these keywords.

    JEL classification:

    • C25 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Discrete Regression and Qualitative Choice Models; Discrete Regressors; Proportions; Probabilities
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C20 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - General

    Statistics

    Access and download statistics

    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:cor:louvco:2016027. 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.

    We have no bibliographic references for this item. You can help adding them by using 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: Alain GILLIS (email available below). General contact details of provider: https://edirc.repec.org/data/coreebe.html .

    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.