IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v187y2020i1d10.1007_s10957-020-01740-8.html
   My bibliography  Save this article

A Gradient Sampling Method Based on Ideal Direction for Solving Nonsmooth Optimization Problems

Author

Listed:
  • Morteza Maleknia

    (Amirkabir University of Technology)

  • Mostafa Shamsi

    (Amirkabir University of Technology)

Abstract

In this paper, a modification to the original gradient sampling method for minimizing nonsmooth nonconvex functions is presented. One computational component in the gradient sampling method is the need to solve a quadratic optimization problem at each iteration, which may result in a time-consuming process, especially for large-scale objectives. To resolve this difficulty, this study proposes a new descent direction, for which there is no need to consider any quadratic or linear subproblem. It is shown that this direction satisfies the Armijo step size condition. We also prove that under proposed modifications, the global convergence of the gradient sampling method is preserved. Moreover, under some moderate assumptions, an upper bound for the number of serious iterations is presented. Using this upper bound, we develop a different strategy to study the convergence of the method. We also demonstrate the efficiency of the proposed method using small-, medium- and large-scale problems in our numerical experiments.

Suggested Citation

  • Morteza Maleknia & Mostafa Shamsi, 2020. "A Gradient Sampling Method Based on Ideal Direction for Solving Nonsmooth Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 181-204, October.
    • Handle: RePEc:spr:joptap:v:187:y:2020:i:1:d:10.1007_s10957-020-01740-8
    DOI: 10.1007/s10957-020-01740-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-020-01740-8
    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/s10957-020-01740-8?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. Elias Salomão Helou & Sandra A. Santos & Lucas E. A. Simões, 2017. "On the Local Convergence Analysis of the Gradient Sampling Method for Finite Max-Functions," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 137-157, October.
    2. Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Introduction to Nonsmooth Optimization," Springer Books, Springer, edition 127, number 978-3-319-08114-4, December.
    3. Elias S. Helou & Sandra A. Santos & Lucas E. A. Simões, 2018. "A fast gradient and function sampling method for finite-max functions," Computational Optimization and Applications, Springer, vol. 71(3), pages 673-717, December.
    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. Maleknia, Morteza & Soleimani-damaneh, Majid, 2024. "An effective subgradient algorithm via Mifflin’s line search for nonsmooth nonconvex multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 319(2), pages 505-516.

    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. M. Maleknia & M. Shamsi, 2020. "A new method based on the proximal bundle idea and gradient sampling technique for minimizing nonsmooth convex functions," Computational Optimization and Applications, Springer, vol. 77(2), pages 379-409, November.
    2. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2018. "Minimizing Piecewise-Concave Functions Over Polyhedra," Mathematics of Operations Research, INFORMS, vol. 43(2), pages 580-597, May.
    3. Konstantin Sonntag & Bennet Gebken & Georg Müller & Sebastian Peitz & Stefan Volkwein, 2024. "A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 455-487, October.
    4. Karmitsa, Napsu & Bagirov, Adil M. & Taheri, Sona, 2017. "New diagonal bundle method for clustering problems in large data sets," European Journal of Operational Research, Elsevier, vol. 263(2), pages 367-379.
    5. 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.
    6. Felipe Serrano & Robert Schwarz & Ambros Gleixner, 2020. "On the relation between the extended supporting hyperplane algorithm and Kelley’s cutting plane algorithm," Journal of Global Optimization, Springer, vol. 78(1), pages 161-179, September.
    7. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    8. Tapio Westerlund & Ville-Pekka Eronen & Marko M. Mäkelä, 2018. "On solving generalized convex MINLP problems using supporting hyperplane techniques," Journal of Global Optimization, Springer, vol. 71(4), pages 987-1011, August.
    9. Ville-Pekka Eronen & Jan Kronqvist & Tapio Westerlund & Marko M. Mäkelä & Napsu Karmitsa, 2017. "Method for solving generalized convex nonsmooth mixed-integer nonlinear programming problems," Journal of Global Optimization, Springer, vol. 69(2), pages 443-459, October.
    10. Kaisa Joki & Adil M. Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2017. "A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes," Journal of Global Optimization, Springer, vol. 68(3), pages 501-535, July.
    11. Nader Kanzi & Majid Soleimani-damaneh, 2020. "Characterization of the weakly efficient solutions in nonsmooth quasiconvex multiobjective optimization," Journal of Global Optimization, Springer, vol. 77(3), pages 627-641, July.
    12. Olivier Morand & Kevin Reffett & Suchismita Tarafdar, 2018. "Generalized Envelope Theorems: Applications to Dynamic Programming," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 650-687, March.
    13. Li-Ping Pang & Qi Wu & Jin-He Wang & Qiong Wu, 2020. "A discretization algorithm for nonsmooth convex semi-infinite programming problems based on bundle methods," Computational Optimization and Applications, Springer, vol. 76(1), pages 125-153, May.
    14. H. Apolinário & E. Papa Quiroz & P. Oliveira, 2016. "A scalarization proximal point method for quasiconvex multiobjective minimization," Journal of Global Optimization, Springer, vol. 64(1), pages 79-96, January.
    15. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2022. "Essentials of numerical nonsmooth optimization," Annals of Operations Research, Springer, vol. 314(1), pages 213-253, July.
    16. Javad Koushki & Majid Soleimani-damaneh, 2020. "Characterization of generalized FJ and KKT conditions in nonsmooth nonconvex optimization," Journal of Global Optimization, Springer, vol. 76(2), pages 407-431, February.
    17. Chungen Shen & Xiao Liu, 2021. "Solving nonnegative sparsity-constrained optimization via DC quadratic-piecewise-linear approximations," Journal of Global Optimization, Springer, vol. 81(4), pages 1019-1055, December.
    18. Joki, Kaisa & Bagirov, Adil M. & Karmitsa, Napsu & Mäkelä, Marko M. & Taheri, Sona, 2020. "Clusterwise support vector linear regression," European Journal of Operational Research, Elsevier, vol. 287(1), pages 19-35.
    19. Giorgio Giorgi, 2021. "Some Classical Directional Derivatives and Their Use in Optimization," DEM Working Papers Series 204, University of Pavia, Department of Economics and Management.
    20. Elias S. Helou & Sandra A. Santos & Lucas E. A. Simões, 2018. "A fast gradient and function sampling method for finite-max functions," Computational Optimization and Applications, Springer, vol. 71(3), pages 673-717, December.

    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:187:y:2020:i:1:d:10.1007_s10957-020-01740-8. 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.