Tikhonov regularization of a second order dynamical system with Hessian driven damping

Abstract

We investigate the asymptotic properties of the trajectories generated by a second-order dynamical system with Hessian driven damping and a Tikhonov regularization term in connection with the minimization of a smooth convex function in Hilbert spaces. We obtain fast convergence results for the function values along the trajectories. The Tikhonov regularization term enables the derivation of strong convergence results of the trajectory to the minimizer of the objective function of minimum norm. Keywords Second order dynamical system • Convex optimization • Tikhonov regularization • Fast convergence methods • Hessian-driven damping Mathematics Subject Classification 34G25 • 47J25 • 47H05 • 90C26 • 90C30 • 65K10 The paper is dedicated to Prof. Marco A. López on the occasion of his 70th birthday.

Continuous Newton-like Inertial Dynamics for Monotone Inclusions

“…While preserving the convergence properties of the Nesterov accelerated method, they provide fast convergence to zero of the gradients, and reduce the oscillatory aspects. Several recent studies have been devoted to this subject, see Attouch-Chbani-Fadili-Riahi [7], Boţ-Csetnek-László [25], Kim [29], Lin-Jordan [31], Shi-Du-Jordan-Su [35].…”

Newton-like Inertial Dynamics and Proximal Algorithms Governed by Maximally Monotone Operators

“…While preserving the convergence properties of the Nesterov accelerated method, they provide fast convergence to zero of the gradients, and reduce the oscillatory aspects. Several recent studies have been devoted to this subject, see Attouch-Chbani-Fadili-Riahi [7], Boţ-Csetnek-László [21], Kim [26], Lin-Jordan [27], Shi-Du-Jordan-Su [31].…”

Fast convex optimization via inertial dynamics combining viscous and Hessian-driven damping with time rescaling

“…Our work is part of the active research stream that studies the close link between continuous dissipative dynamical systems and optimization algorithms. In general, the implicit temporal discretization of continuous gradient-based dynamics provides proximal algorithms that benefit from similar asymptotic convergence properties, see [29] for a systematic study in the case of first-order evolution systems, and [5,6,8,12,11,19,20,21] for some recent results concerning second-order evolution equations. The main object of our study is the second-order in time differential equation (IGS) γ,β,bẍ (t) + γ(t)ẋ(t) + β(t)∇ 2 f (x(t))ẋ(t) + b(t)∇f (x(t)) = 0,…”