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

Abstract

AbstractLet H be a real Hilbert space, and f:H→R be a convex twice differentiable function whose solution set argminf 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 x¨(t)+α x˙(t)+δ∇2f(x(t))x˙(t)+β(t)∇f(x(t))+cx(t)=0 where α,c,δ are three positive constants, and the time scale parameter β is a positive nondecreasing function such that limt→+∞β(t)=+∞. Under some assumptions on the parameter β, we will show rapid convergence of values, strong convergence toward the minimum norm element of argminf, 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 β, we set β(t)=tp ln q(t), for (p,q)∈(R+)2∖{(0,0)}, and β(t)=eγtp, for p∈]0,1[ and γ>0. The manuscript concludes with two numerical examples and comments on their performance.KEYWORDS: Damped dynamical systemfast convergenceHessian-driven dampingHilbert spacestrong convergenceTikhonov regularizationMATHEMATICS SUBJECT CLASSIFICATION: 37N4046N1049M3065K0565K1090B5090C25