Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient

Hedy Attouch; Alexandre Cabot; Zaki Chbani; Hassan Riahi

doi:10.3934/eect.2018018

Article Contents

2018, Volume 7, Issue 3: 353-371. Doi: 10.3934/eect.2018018

This issue Previous Article Energy decay for the damped wave equation on an unbounded network Next Article Existence and continuous-discrete asymptotic behaviour for Tikhonov-like dynamical equilibrium systems

Rate of convergence of inertial gradient dynamics with time-dependent viscous damping coefficient

1.
IMAG, Univ. Montpellier, CNRS, Montpellier, France
2.
Institut de Mathématiques de Bourgogne, UMR 5584, Université Bourgogne Franche-Comté, 21000 Dijon, France
3.
Faculty of Sciences Semlalia, Mathematics, Cadi Ayyad university, 40000 Marrakech, Morroco

* Corresponding author
* Corresponding author

Received: December 2017

Revised: April 2018

Published: July 2018

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Abstract

In a Hilbert space $\mathcal H$ , we study the convergence properties when $t→+∞$ of the trajectories of the second-order differential equation

$\begin{equation*}\ddot{x}(t) + γ(t) \dot{x}(t) + \nabla Φ (x(t)) = 0, \;\;\;\;\;\;\;\;{{\rm (IGS)}_{γ}}\end{equation*}$

where $\nablaΦ$ is the gradient of a convex continuously differentiable function $Φ: \mathcal H→\mathbb R$ , and $γ(t)$ is a time-dependent positive viscous damping coefficient. This study aims to offer a unifying vision on the subject, and to complement the article by Attouch and Cabot (J. Diff. Equations, 2017). We obtain convergence rates for the values $Φ(x(t))-{\rm{inf}}_\mathcal{H} Φ$ and the velocities under general conditions involving only $γ(·)$ and its derivatives. In particular, in the case $γ(t) = \frac{α}{t}$ , which is directly connected to the Nesterov accelerated gradient method, our approach allows us to cover all the positive values of $α$ , including the subcritical case $α<3$ . Our approach is based on the introduction of a new class of Lyapunov functions.

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Mathematics Subject Classification: 37N40, 49M30, 65K05, 65K10, 90C25.

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