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

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.