Analyzing the speed of convergence in nonsmooth optimization via the Goldstein subdifferential with application to descent methods

Abstract

The Goldstein $\epsilon$-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 $(\epsilon ,\delta )$-critical points, which are points in which the element with the smallest norm in the $\epsilon$-subdifferential has norm at most $\delta$. To obtain points that are critical in the classical sense, $\epsilon$ and $\delta$ must vanish. In this article, we analyze at which speed the distance of $(\epsilon ,\delta )$-critical points to the minimum vanishes with respect to $\epsilon$ and $\delta$. 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.

arXiv commentThis preprint has not undergone peer review or any post-submission improvements or corrections. The Version of Record of this article is published in the Journal of Optimization Theory and Applications, and is available online at https://doi.org/10.1007/s10957-025-02748-8

References in corpus (8)

• An efficient descent method for locally Lipschitz multiobjective optimization problems
• Can We Find Near-Approximately-Stationary Points of Nonsmooth Nonconvex Functions?
• Survey Descent: A Multipoint Generalization of Gradient Descent for Nonsmooth Optimization
• Exploiting higher-order derivatives in convex optimization methods
• A note on the convergence of deterministic gradient sampling in nonsmooth optimization
• Using second-order information in gradient sampling methods for nonsmooth optimization
• Convergence of Descent Optimization Algorithms under Polyak-Łojasiewicz-Kurdyka Conditions
• Lipschitz minimization and the Goldstein modulus