On the strong convergence of the trajectories of a Tikhonov regularized second order dynamical system with asymptotically vanishing damping

TL;DR

This paper studies a second order dynamical system with vanishing damping and Tikhonov regularization, showing convergence of trajectories and objective function values to the minimum, with improved results on strong convergence to minimal norm solutions.

Contribution

It extends recent results by establishing strong convergence of trajectories to minimal norm solutions in a Tikhonov regularized second order system with vanishing damping.

Findings

Trajectory converges weakly to a minimizer.

Objective function value converges rapidly to the global minimum.

Velocities tend to zero quickly.

Abstract

This paper deals with a second order dynamical system with vanishing damping that contains a Tikhonov regularization term, in connection to the minimization problem of a convex Fr\'echet differentiable function $g$. We show that for appropriate Tikhonov regularization parameters the value of the objective function in a generated trajectory converges fast to the global minimum of the objective function and a trajectory generated by the dynamical system converges weakly to a minimizer of the objective function. We also obtain the fast convergence of the velocities towards zero and some integral estimates. Nevertheless, our main goal is to extend and improve some recent results obtained in \cite{ABCR} and \cite{AL-nemkoz} concerning the strong convergence of the generated trajectories to an element of minimal norm from the $\argmin$ set of the objective function $g$. Our analysis also reveals that the damping coefficient and the Tikhonov regularization coefficient are strongly correlated.