Dynamical system related to primal-dual splitting projection methods

TL;DR

This paper introduces a dynamical system framework for solving monotone inclusion problems, proving convergence to solutions and deriving an optimal algorithm via discretization.

Contribution

It presents a novel dynamical system approach for primal-dual problems and establishes strong convergence results, leading to an efficient approximation algorithm.

Findings

Trajectories converge strongly to primal-dual solutions.

Existence, uniqueness, and extendability of solutions are established.

Discretization yields an optimal algorithm for coupled monotone inclusions.

Abstract

We introduce a dynamical system to the problem of finding zeros of the sum of two maximally monotone operators. We investigate the existence, uniqueness and extendability of solutions to this dynamical system in a Hilbert space. We prove that the trajectories of the proposed dynamical system converge strongly to a primal-dual solution of the considered problem. Under explicit time discretization of the dynamical system we obtain the best approximation algorithm for solving coupled monotone inclusion problem.