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

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  • 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.

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


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