Second order splitting dynamics with vanishing damping for additively structured monotone inclusions

TL;DR

This paper analyzes a second order dynamical system with vanishing damping for solving structured monotone inclusions, demonstrating weak convergence and fast rates, with applications to convex optimization problems.

Contribution

It introduces a new second order splitting method with vanishing damping for monotone inclusions, ensuring convergence and efficiency in solving structured optimization problems.

Findings

Weak convergence of trajectories to zeros of the sum operator

Fast convergence rates for optimization problems

Numerical experiments confirming theoretical results

Abstract

In the framework of a real Hilbert space, we address the problem of finding the zeros of the sum of a maximally monotone operator $A$ and a cocoercive operator $B$. We study the asymptotic behaviour of the trajectories generated by a second order equation with vanishing damping, attached to this problem, and governed by a time-dependent forward-backward-type operator. This is a splitting system, as it only requires forward evaluations of $B$ and backward evaluations of $A$. A proper tuning of the system parameters ensures the weak convergence of the trajectories to the set of zeros of $A + B$, as well as fast convergence of the velocities towards zero. A particular case of our system allows to derive fast convergence rates for the problem of minimizing the sum of a proper, convex and lower semicontinuous function and a smooth and convex function with Lipschitz continuous gradient. We illustrate the theoretical outcomes by numerical experiments.