Second order forward-backward dynamical systems for monotone inclusion problems

TL;DR

This paper introduces second order dynamical systems for solving monotone inclusion problems, proving existence, uniqueness, and convergence of trajectories, with applications to convex optimization and convergence rate analysis.

Contribution

It develops a novel second order dynamical framework for monotone inclusions, including convergence results and rate analysis for convex optimization applications.

Findings

Trajectories exist uniquely and converge weakly to zeros of the operator.

Ergodic trajectories of the function converge at a rate of O(1/t).

Framework encompasses problems involving sums of monotone operators and convex functions.

Abstract

We begin by considering second order dynamical systems of the from $\ddot x(t) + \gamma(t)\dot x(t) + \lambda(t)B(x(t))=0$, where $B: {\cal H}\rightarrow{\cal H}$ is a cocoercive operator defined on a real Hilbert space ${\cal H}$, $\lambda:[0,+\infty)\rightarrow [0,+\infty)$ is a relaxation function and $\gamma:[0,+\infty)\rightarrow [0,+\infty)$ a damping function, both depending on time. For the generated trajectories, we show existence and uniqueness of the generated trajectories as well as their weak asymptotic convergence to a zero of the operator $B$. The framework allows to address from similar perspectives second order dynamical systems associated with the problem of finding zeros of the sum of a maximally monotone operator and a cocoercive one. This captures as particular case the minimization of the sum of a nonsmooth convex function with a smooth convex one. Furthermore, we prove that when $B$ is the gradient of a smooth convex function the value of the latter converges along the ergodic trajectory to its minimal value with a rate of ${\cal O}(1/t)$.