A forward-backward dynamical approach to the minimization of the sum of a nonsmooth convex with a smooth nonconvex function

TL;DR

This paper introduces a dynamical system approach combining forward-backward methods to minimize a sum of a convex nonsmooth and a smooth nonconvex function, proving convergence to critical points.

Contribution

It formulates a novel implicit dynamical system for such minimization problems and establishes convergence and rate results under the Kurdyka-lasiewicz property.

Findings

Trajectory converges to critical points under certain conditions.

Convergence rates depend on the lasiewicz exponent.

Provides theoretical guarantees for the proposed dynamical approach.

Abstract

We address the minimization of the sum of a proper, convex and lower semicontinuous with a (possibly nonconvex) smooth function from the perspective of an implicit dynamical system of forward-backward type. The latter is formulated by means of the gradient of the smooth function and of the proximal point operator of the nonsmooth one. The trajectory generated by the dynamical system is proved to asymptotically converge to a critical point of the objective, provided a regularization of the latter satisfies the Kurdyka-\L{}ojasiewicz property. Convergence rates for the trajectory in terms of the \L{}ojasiewicz exponent of the regularized objective function are also provided.