RATE OF CONVERGENCE OF INERTIAL GRADIENT DYNAMICS WITH TIME-DEPENDENT VISCOUS DAMPING COEFFICIENT

Abstract : In a Hilbert space $\mathcal H$, we study the convergence properties when $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \mbox{(IGS)}_{\gamma} \quad \quad \ddot{x}(t) + \gamma(t) \dot{x}(t) + \nabla \Phi (x(t))=0, \end{equation*} where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, and $\gamma(t)$ 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 $\Phi(x(t))-\inf_{\mathcal H} \Phi$ and the velocities under general conditions involving only $\gamma (\cdot)$ and its derivatives. In particular, in the case $\gamma(t) = \frac{\alpha}{t}$, which is directly connected to the Nesterov accelerated gradient method, our approach allows to cover all the positive values of $\alpha$, including the subcritical case $\alpha <3$. Our approach is based on the introduction of a new class of Lyapunov functions.
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Pré-publication, Document de travail
2018
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Soumis le : samedi 7 avril 2018 - 12:58:35
Dernière modification le : mardi 10 avril 2018 - 06:34:54

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  • HAL Id : hal-01660062, version 2

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Hedy Attouch, Alexandre Cabot, Zaki Chbani, Hassan Riahi. RATE OF CONVERGENCE OF INERTIAL GRADIENT DYNAMICS WITH TIME-DEPENDENT VISCOUS DAMPING COEFFICIENT. 2018. 〈hal-01660062v2〉

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