A forward-backward-forward differential equation and its asymptotic properties

TL;DR

This paper introduces a new forward-backward-forward differential equation approach for finding zeros of sums of monotone operators, proving convergence properties and rates in Hilbert spaces.

Contribution

It develops an implicit dynamical system with nonconstant parameters for solving monotone inclusion problems, establishing existence, uniqueness, and convergence results.

Findings

Weak convergence of trajectories is proven.

Strong convergence with exponential rate under monotonicity.

Convergence rate for objective function in convex minimization.

Abstract

In this paper, we approach the problem of finding the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space via an implicit forward-backward-forward dynamical system with nonconstant relaxation parameters and stepsizes of the resolvents. Besides proving existence and uniqueness of strong global solutions for the differential equation under consideration, we show weak convergence of the generated trajectories and, under strong monotonicity assumptions, strong convergence with exponential rate. In the particular setting of minimizing the sum of a proper, convex and lower semicontinuous function with a smooth convex one, we provide a rate for the convergence of the objective function along the ergodic trajectory to its minimum value.