Approaching nonsmooth nonconvex optimization problems through first order dynamical systems with hidden acceleration and Hessian driven damping terms

TL;DR

This paper analyzes a first-order dynamical system combining proximal-gradient steps with hidden acceleration and damping terms, showing convergence to critical points of a nonsmooth, nonconvex optimization problem under Kurdyka-{}ojasiewicz conditions.

Contribution

It introduces a novel dynamical system framework for nonsmooth nonconvex optimization with acceleration and damping, proving convergence to critical points.

Findings

Trajectories approach the set of critical points.

Convergence rates depend on the Kurdyka-{}ojasiewicz exponent.

Results apply under mild regularity assumptions.

Abstract

In this paper we carry out an asymptotic analysis of the proximal-gradient dynamical system \begin{equation*}\left\{ \begin{array}{ll} \dot x(t) +x(t) = \prox_{\gamma f}\big[x(t)-\gamma\nabla\Phi(x(t))-ax(t)-by(t)\big],\\ \dot y(t)+ax(t)+by(t)=0 \end{array}\right.\end{equation*} where $f$ is a proper, convex and lower semicontinuous function, $\Phi$ a possibly nonconvex smooth function and $\gamma, a$ and $b$ are positive real numbers. We show that the generated trajectories approach the set of critical points of $f+\Phi$, here understood as zeros of its limiting subdifferential, under the premise that a regularization of this sum function satisfies the Kurdyka-\L{}ojasiewicz property. We also establish convergence rates for the trajectories, formulated in terms of the \L{}ojasiewicz exponent of the considered regularization function.