Tikhonov regularized second-order dynamical systems with Hessian-driven damping for solving convex optimization problems

Abstract: This paper deals with a Tikhonov regularized second-order 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 the fast convergence rate of the function value and the strong convergence of the trajectory towards the minimum norm solution of the convex optimization problem can be achieved simultaneously. Finally, we present numerical experiments to illustrate the obtained results.