Convergence analysis of a Tikhonov regularized inertial dynamical system and algorithm for convex optimization problems

TL;DR

This paper analyzes a Tikhonov regularized inertial dynamical system with time scaling and damping, providing convergence results and an associated algorithm for convex optimization, supported by numerical experiments.

Contribution

It introduces a novel dynamical system with specific damping and regularization techniques, and derives convergence properties and an algorithm for convex optimization.

Findings

Fast convergence of function values along trajectories.

Weak convergence to a minimizer of the convex problem.

Simultaneous fast convergence and strong convergence to minimum norm solution.

Abstract

This paper deals with a Tikhonov regularized second-order inertial dynamical system that incorporates time scaling, asymptotically vanishing damping and Hessian-driven damping for solving convex optimization problems. Under appropriate setting of the parameters, we first obtain fast convergence results of the function value along the trajectory generated by the dynamical system. Then, we show that the trajectory generated by the dynamical system converges weakly to a minimizer of the convex optimization problem. We also demonstrate that, by properly tuning these parameters, both fast convergence rates of the function value and strong convergence of the trajectory towards the minimum norm solution of the convex optimization problem can be achieved simultaneously. Furthermore, we study convergence properties of an inertial proximal gradient algorithm obtained by the temporal discretization of the dynamical system. Finally, we present numerical experiments to illustrate the obtained results.