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

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  • 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
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    References listed on IDEAS

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    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, February.
    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.
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