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

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