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Subgradient methods comprise a fundamental class of nonsmooth optimizationalgorithms. Classical results show that certain subgradient methods convergesublinearly for general Lipschitz convex functions and converge linearly forconvex functions that grow sharply away from solutions. Recent work hasmoreover extended these results to certain nonconvex problems. In this work weseek to improve the complexity of these algorithms, asking: is it possible todesign a superlinearly convergent subgradient method? We provide a positiveanswer to this question for a broad class of sharp semismooth functions.

A superlinearly convergent subgradient method for sharp semismooth problems