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Analyzing the speed of convergence in nonsmooth optimization via the Goldstein subdifferential with application to descent methods
Bennet Gebken
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The Goldstein 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 (ε,δ)-critical points, which are points in which the element with the smallest norm in the ε-subdifferential has norm at most δ. To obtain points that are critical in the classical sense, ε and δ must vanish. In this article, we analyze at which speed the distance of (ε,δ)-critical points to the minimum vanishes with respect to ε and 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.
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