Strong asymptotic convergence of a slowly damped inertial primal-dual dynamical system controlled by a Tikhonov regularization term

TL;DR

This paper introduces a novel inertial primal-dual dynamical system with Tikhonov regularization for linearly constrained convex optimization, proving strong convergence and providing convergence rates, supported by numerical experiments.

Contribution

It presents a new inertial primal-dual system with Tikhonov regularization, demonstrating strong asymptotic convergence in Hilbert spaces, which was not previously established.

Findings

Strong asymptotic convergence to minimal norm solution

Convergence rate results for primal-dual gap and residuals

Numerical experiments confirming theoretical results

Abstract

We propose a slowly damped inertial primal-dual dynamical system controlled by a Tikhonov regularization term, where the inertial term is introduced only for the primal variable, for the linearly constrained convex optimization problem in a Hilbert space. Under mild conditions on the underlying parameters, by a Lyapunov analysis approach, we prove the strong asymptotic convergence of the trajectory of the proposed dynamic to the minimal norm element of the primal-dual solution set of the problem, along with convergence rate results for the primal-dual gap, the objective residual and the feasibility violation. We perform some numerical experiments to illustrate the theoretical findings.