A Newton-Like Dynamical System for Nonsmooth and Nonconvex Optimization

TL;DR

This paper introduces a Newton-like dynamical system designed for nonsmooth, nonconvex optimization problems, extending classical methods with new theoretical insights into convergence and stability.

Contribution

It develops a novel continuous dynamical system that generalizes Newton's method for nonsmooth, nonconvex functions, analyzing its solutions and convergence properties.

Findings

Existence and uniqueness of solutions established

Convergence conditions under metric subregularity and Kurdyka-Lojasiewicz inequality

Convergence rates characterized for different scenarios

Abstract

This work investigates a dynamical system functioning as a nonsmooth adaptation of the continuous Newton method, aimed at minimizing the sum of a primal lower-regular and a locally Lipschitz function, both potentially nonsmooth. The classical Newton method's second-order information is extended by incorporating the graphical derivative of a locally Lipschitz mapping. Specifically, we analyze the existence and uniqueness of solutions, along with the asymptotic behavior of the system's trajectories. Conditions for convergence and respective convergence rates are established under two distinct scenarios: strong metric subregularity and satisfaction of the Kurdyka-Lojasiewicz inequality.