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

Bennet Gebken ()
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Bennet Gebken: Technical University of Munich

Journal of Optimization Theory and Applications, 2025, vol. 206, issue 3, No 9, 38 pages

Abstract: 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.

Keywords: Nonsmooth optimization; Nonsmooth analysis; Nonconvex optimization; 90C30; 90C56; 49J52 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s10957-025-02748-8

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