IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v144y2010i1d10.1007_s10957-009-9584-6.html
   My bibliography  Save this article

Improved Convergence Result for the Discrete Gradient and Secant Methods for Nonsmooth Optimization

Author

Listed:
  • K. C. Kiwiel

    (Polish Academy of Sciences)

Abstract

We study a generalization of the nonderivative discrete gradient method of Bagirov et al. for minimizing a locally Lipschitz function f on ℝ n . We strengthen the existing convergence result for this method by showing that it either drives the f-values to −∞ or each of its cluster points is Clarke stationary for f, without requiring the compactness of the level sets of f. Our generalization is an approximate bundle method, which also subsumes the secant method of Bagirov et al.

Suggested Citation

  • K. C. Kiwiel, 2010. "Improved Convergence Result for the Discrete Gradient and Secant Methods for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 144(1), pages 69-75, January.
    • Handle: RePEc:spr:joptap:v:144:y:2010:i:1:d:10.1007_s10957-009-9584-6
    DOI: 10.1007/s10957-009-9584-6
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-009-9584-6
    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/s10957-009-9584-6?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. A. M. Bagirov & B. Karasözen & M. Sezer, 2008. "Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 317-334, May.
    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. Adil M. Bagirov & Julien Ugon & Hijran G. Mirzayeva, 2015. "Nonsmooth Optimization Algorithm for Solving Clusterwise Linear Regression Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 755-780, March.
    2. Adil Bagirov & Asef Ganjehlou, 2008. "An approximate subgradient algorithm for unconstrained nonsmooth, nonconvex optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(2), pages 187-206, April.
    3. A. Bagirov & B. Ordin & G. Ozturk & A. Xavier, 2015. "An incremental clustering algorithm based on hyperbolic smoothing," Computational Optimization and Applications, Springer, vol. 61(1), pages 219-241, May.
    4. Chad Davis & Warren Hare, 2013. "Exploiting Known Structures to Approximate Normal Cones," Mathematics of Operations Research, INFORMS, vol. 38(4), pages 665-681, November.
    5. Adil Bagirov & Julien Ugon & Dean Webb & Gurkan Ozturk & Refail Kasimbeyli, 2013. "A novel piecewise linear classifier based on polyhedral conic and max–min separabilities," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(1), pages 3-24, April.
    6. Gaudioso, Manlio & Giallombardo, Giovanni & Mukhametzhanov, Marat, 2018. "Numerical infinitesimals in a variable metric method for convex nonsmooth optimization," Applied Mathematics and Computation, Elsevier, vol. 318(C), pages 312-320.
    7. Napsu Karmitsa, 2016. "Testing Different Nonsmooth Formulations of the Lennard–Jones Potential in Atomic Clustering Problems," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 316-335, October.
    8. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    9. M. V. Dolgopolik, 2018. "A convergence analysis of the method of codifferential descent," Computational Optimization and Applications, Springer, vol. 71(3), pages 879-913, December.
    10. A. Bagirov & J. Ugon, 2011. "Codifferential method for minimizing nonsmooth DC functions," Journal of Global Optimization, Springer, vol. 50(1), pages 3-22, May.
    11. Shuai Liu, 2019. "A simple version of bundle method with linear programming," Computational Optimization and Applications, Springer, vol. 72(2), pages 391-412, March.
    12. M. Golestani & H. Sadeghi & Y. Tavan, 2018. "Nonsmooth Multiobjective Problems and Generalized Vector Variational Inequalities Using Quasi-Efficiency," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 896-916, December.
    13. A. M. Bagirov & L. Jin & N. Karmitsa & A. Al Nuaimat & N. Sultanova, 2013. "Subgradient Method for Nonconvex Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 157(2), pages 416-435, May.
    14. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2022. "Essentials of numerical nonsmooth optimization," Annals of Operations Research, Springer, vol. 314(1), pages 213-253, July.
    15. Gonglin Yuan & Zehong Meng & Yong Li, 2016. "A Modified Hestenes and Stiefel Conjugate Gradient Algorithm for Large-Scale Nonsmooth Minimizations and Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 129-152, January.
    16. Nezam Mahdavi-Amiri & Rohollah Yousefpour, 2012. "An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 180-195, October.
    17. Kalyanmoy Deb & Shivam Gupta & Joydeep Dutta & Bhoomija Ranjan, 2013. "Solving dual problems using a coevolutionary optimization algorithm," Journal of Global Optimization, Springer, vol. 57(3), pages 891-933, November.

    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:144:y:2010:i:1:d:10.1007_s10957-009-9584-6. 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.