Approaching nonsmooth nonconvex minimization through second order proximal-gradient dynamical systems

TL;DR

This paper studies the long-term behavior of second-order dynamical systems used for minimizing nonsmooth, possibly nonconvex functions, establishing convergence to critical points under certain regularity conditions and providing convergence rates.

Contribution

It introduces a second-order proximal-gradient dynamical system framework for nonsmooth nonconvex minimization and proves convergence to critical points with rates based on the Kurdyka-{}ojasiewicz property.

Findings

Trajectory converges to critical points under Kurdyka-{}ojasiewicz condition.

Provides explicit convergence rates depending on the {}ojasiewicz exponent.

Extends analysis to nonsmooth, nonconvex optimization problems.

Abstract

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system of proximal-gradient type stated in connection with the minimization of the sum of a nonsmooth convex and a (possibly nonconvex) smooth function. The convergence of the generated trajectory to a critical point of the objective is ensured provided a regularization of the objective function satisfies the Kurdyka-\L{}ojasiewicz property. We also provide convergence rates for the trajectory formulated in terms of the \L{}ojasiewicz exponent.