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.
