Convergence rates for forward-backward dynamical systems associated with strongly monotone inclusions

TL;DR

This paper analyzes the exponential convergence rates of forward-backward dynamical systems in Hilbert spaces for strongly monotone inclusions and strongly convex optimization problems, providing theoretical guarantees.

Contribution

It establishes exponential convergence rates for trajectories of dynamical systems solving strongly monotone inclusions and strongly convex optimization problems, extending existing results.

Findings

Trajectories strongly converge with exponential rate when the operator is strongly monotone.

Convergence rates are derived for minimization problems involving strongly convex functions.

Function values converge exponentially to the minimum in the strongly convex case.

Abstract

We investigate the convergence rates of the trajectories generated by implicit first and second order dynamical systems associated to the determination of the zeros of the sum of a maximally monotone operator and a monotone and Lipschitz continuous one in a real Hilbert space. We show that these trajectories strongly converge with exponential rate to a zero of the sum, provided the latter is strongly monotone. We derive from here convergence rates for the trajectories generated by dynamical systems associated to the minimization of the sum of a proper, convex and lower semicontinuous function with a smooth convex one provided the objective function fulfills a strong convexity assumption. In the particular case of minimizing a smooth and strongly convex function, we prove that its values converge along the trajectory to its minimum value with exponential rate, too.