A superlinearly convergent subgradient method for sharp semismooth problems

TL;DR

This paper introduces a subgradient method that achieves superlinear convergence for a broad class of sharp semismooth functions, improving the efficiency of nonsmooth optimization algorithms.

Contribution

It presents the first superlinearly convergent subgradient method applicable to sharp semismooth problems, extending the capabilities of existing nonsmooth optimization techniques.

Findings

Achieves superlinear convergence for sharp semismooth functions

Extends convergence results beyond convex functions

Provides theoretical foundation for faster nonsmooth optimization algorithms

Abstract

Subgradient methods comprise a fundamental class of nonsmooth optimization algorithms. Classical results show that certain subgradient methods converge sublinearly for general Lipschitz convex functions and converge linearly for convex functions that grow sharply away from solutions. Recent work has moreover extended these results to certain nonconvex problems. In this work we seek to improve the complexity of these algorithms, asking: is it possible to design a superlinearly convergent subgradient method? We provide a positive answer to this question for a broad class of sharp semismooth functions.