Fast convergence rates and trajectory convergence of a Tikhonov regularized inertial primal\mbox{-}dual dynamical system with time scaling and vanishing damping

TL;DR

This paper introduces a Tikhonov regularized inertial primal-dual dynamical system with time scaling and vanishing damping, achieving fast convergence rates and strong trajectory convergence for linearly constrained convex optimization problems.

Contribution

It develops a novel dynamical system with specific decay conditions that guarantees both fast convergence and strong solution convergence, extending prior methods.

Findings

Fast convergence rates for primal-dual gap and objective residual.

Weak convergence of trajectories to primal-dual solutions.

Strong convergence to minimal norm solutions under certain conditions.

Abstract

A Tikhonov regularized inertial primal\mbox{-}dual dynamical system with time scaling and vanishing damping is proposed for solving a linearly constrained convex optimization problem in Hilbert spaces. The system under consideration consists of two coupled second order differential equations and its convergence properties depend upon the decaying speed of the product of the time scaling parameter and the Tikhonov regularization parameter (named the rescaled regularization parameter) to zero. When the rescaled regularization parameter converges rapidly to zero, the system enjoys fast convergence rates of the primal-dual gap, the feasibility violation, the objective residual, and the gradient norm of the objective function along the trajectory, and the weak convergence of the trajectory to a primal-dual solution of the linearly constrained convex optimization problem. When the rescaled regularization parameter converges slowly to zero, the generated primal trajectory converges strongly to the minimal norm solution of the problem under suitable conditions. Finally, numerical experiments are performed to illustrate the theoretical findings.