Strong Convergence of Trajectories via Inertial Dynamics Combining Hessian-Driven Damping and Tikhonov Regularization for General Convex Minimizations

Abstract

Let be a real Hilbert space, and be a convex twice differentiable function whose solution set is nonempty. We investigate the long time behavior of the trajectories of the vanishing damped dynamical system with Tikhonov regularizing term and Hessian-driven damping where are three positive constants, and the time scale parameter β is a positive nondecreasing function such that . Under some assumptions on the parameter , we will show rapid convergence of values, strong convergence toward the minimum norm element of , and rapid convergence of the gradients toward zero. Note that the Hessian-driven damping significantly reduces the oscillatory aspects, and the time scale parameter β improves the rate of convergences mentioned above. As particular cases of , we set , for , and , for and . The manuscript concludes with two numerical examples and comments on their performance.