Solving maximally comonotone inclusion problems via an implicit Newton-like inertial dynamical system and its discretization

TL;DR

This paper introduces a novel inertial dynamical system and its discretization to solve maximally comonotone inclusion problems, demonstrating fast convergence and weak trajectory convergence in a Hilbert space.

Contribution

It develops a new implicit Newton-like inertial system and a discretized algorithm with proven convergence properties for solving maximally comonotone inclusions.

Findings

Weak convergence of the trajectory to a zero of the operator

Fast convergence rates of the discretized algorithm

Numerical experiments confirming theoretical results

Abstract

This paper deals with an implicit Newton-like inertial dynamical system governed by a maximally comonotone inclusion problem in a Hilbert space. Under suitable conditions, we establish not only pointwise estimates and integral estimates for the velocity and the value of the associated Yosida regularization operator along the trajectory of the system, but also the weak convergence of the trajectory to a zero of the maximally comonotone operator. Moreover, a new inertial algorithm is developed via a time discretization of the proposed system. Our analysis reveals that the resulting discrete algorithm exhibits fast convergence properties matching the ones of the continuous time counterpart. Finally, the theoretical results are illustrated by numerical experiments.