Variable metric backward-forward dynamical systems for monotone inclusion problems

TL;DR

This paper introduces variable metric backward-forward dynamical systems for monotone inclusion and convex minimization, proving convergence properties and stability, with numerical examples demonstrating effectiveness.

Contribution

It develops a new class of variable metric dynamical systems related to forward-backward methods, with convergence and stability analysis in Hilbert spaces.

Findings

Existence and uniqueness of trajectories

Weak and strong convergence results

Global exponential stability of equilibrium points

Abstract

This paper investigates first-order variable metric backward forward dynamical systems associated with monotone inclusion and convex minimization problems in real Hilbert space. The operators are chosen so that the backward-forward dynamical system is closely related to the forward-backward dynamical system and has the same computational complexity. We show existence, uniqueness, and weak asymptotic convergence of the generated trajectories and strong convergence if one of the operators is uniformly monotone. We also establish that an equilibrium point of the trajectory is globally exponentially stable and monotone attractor. As a particular case, we explore similar perspectives of the trajectories generated by a dynamical system related to the minimization of the sum of a nonsmooth convex and a smooth convex function. Numerical examples are given to illustrate the convergence of trajectories.