A third order dynamical system for generalized monotone equation

TL;DR

This paper introduces a third order dynamical system for solving nonlinear equations in Hilbert spaces, achieving faster convergence rates than previous second order methods, with applications to convex optimization and monotone inclusion problems.

Contribution

The paper proposes a novel third order dynamical system that guarantees convergence and improves convergence rates for solving monotone equations and convex optimization problems.

Findings

Exponential convergence rate $e^{-2t}$ for strongly monotone operators.

Fast $ ext{O}(1/t^3)$ convergence rate for objective values in convex optimization.

Applicability to splitting monotone inclusion problems.

Abstract

We propose a third order dynamical system for solving a nonlinear equation in Hilbert spaces where the operator is cocoercive with respect to the solutions set. Under mild conditions on the parameters, we establish the existence and uniqueness of the generated trajectories as well as its asymptotic convergence to a solution of the equation. When the operator is strongly monotone with respect to the solutions set, we deliver an exponential convergence rate of $e^{-2t}$, which is significantly faster than the known results of second order dynamical systems. In particular, for convex optimization problems, the proposed dynamical system provides a fast convergence rate of {\bf $\mathcal{O}(\frac{1}{t^3})$} for the objective values. In addition, we discuss the applications of the proposed dynamical system to several splitting monotone inclusion problems.