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Analyzing the Speed of Convergence in Nonsmooth Optimization via the Goldstein subdifferential

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  • Bennet Gebken

    (Technical University of Munich)

Abstract

The Goldstein $$\varepsilon $$ ε -subdifferential is a relaxed version of the Clarke subdifferential which has recently appeared in several algorithms for nonsmooth optimization. With it comes the notion of $$(\varepsilon ,\delta )$$ ( ε , δ ) -critical points, which are points in which the element with the smallest norm in the $$\varepsilon $$ ε -subdifferential has norm at most $$\delta $$ δ . To obtain points that are critical in the classical sense, $$\varepsilon $$ ε and $$\delta $$ δ must vanish. In this article, we analyze at which speed the distance of $$(\varepsilon ,\delta )$$ ( ε , δ ) -critical points to the minimum vanishes with respect to $$\varepsilon $$ ε and $$\delta $$ δ . Afterwards, we apply our results to gradient sampling methods and perform numerical experiments. Throughout the article, we put a special emphasis on supporting the theoretical results with simple examples that visualize them.

Suggested Citation

  • Bennet Gebken, 2025. "Analyzing the Speed of Convergence in Nonsmooth Optimization via the Goldstein subdifferential," Journal of Optimization Theory and Applications, Springer, vol. 206(3), pages 1-38, September.
    • Handle: RePEc:spr:joptap:v:206:y:2025:i:3:d:10.1007_s10957-025-02748-8
    DOI: 10.1007/s10957-025-02748-8
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    References listed on IDEAS

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    1. Bennet Gebken, 2024. "A note on the convergence of deterministic gradient sampling in nonsmooth optimization," Computational Optimization and Applications, Springer, vol. 88(1), pages 151-165, May.
    2. Defeng Sun & Jie Sun, 2008. "Löwner's Operator and Spectral Functions in Euclidean Jordan Algebras," Mathematics of Operations Research, INFORMS, vol. 33(2), pages 421-445, May.
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    4. 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.
    5. Robert Mifflin, 1977. "An Algorithm for Constrained Optimization with Semismooth Functions," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 191-207, May.
    6. Bennet Gebken & Sebastian Peitz, 2021. "An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 696-723, March.
    7. Liqun Qi, 1993. "Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 227-244, February.
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