Asymptotic stabilization of inertial gradient dynamics with time-dependent viscosity

Abstract

International audienceIn a Hilbert space $\mathcal H$, we study the asymptotic behaviour, as time variable $t$ goes to $+\infty$, of nonautonomous gradient-like inertial dynamics, with a time-dependent viscosity coefficient. Given $\Phi: \mathcal H \rightarrow \mathbb R$ a convex differentiable function, $\gamma (\cdot)$ a time-dependent positive damping term, we consider the second-order differential equation $$\ddot{x}(t) + \gamma (t) \dot{x}(t) + \nabla \Phi (x(t)) = 0. $$ This system plays a central role in mechanics and physics in the asymptotic stabilization of nonlinear oscillators. Its importance in optimization was recently put to the fore by Su, Boyd, and Candès. They have shown that in the particular case $\gamma(t) = \frac{3}{t}$, this is a continuous version of the fast gradient method initiated by Nesterov, with $\Phi(x(t))-\min_{\mathcal H} \Phi = \mathcal O (\frac{1}{t^2})$ as $t\to +\infty$ in the worst case. Recently, in the case $\gamma(t) = \frac{\alpha}{t}$ with $\alpha >3$, Attouch and Peypouquet have improved this result by showing the convergence of the trajectories to optimal solutions, and $\Phi(x(t))-\min_{\mathcal H} \Phi = o (\frac{1}{t^2})$ as $t\to +\infty$. For these questions, and the design of fast optimization methods, the tuning of the damping parameter $\gamma(t)$ is a subtle question, which we deal with in this paper in general. We obtain convergence rates for the values, and convergence results of the trajectories under general conditions on $\gamma(\cdot)$ which unify, and often improve the results already present in the literature. We complement these results by showing that they are robust with respect to perturbations

Convergence of damped inertial dynamics governed by regularized maximally monotone operators

“…When the operator is the subdifferential of a closed convex proper function, we estimate the rate of convergence of the values. These results are in line with the recent articles by Attouch-Cabot [3], and Attouch-Peypouquet [8]. In this last paper, the authors considered the case γ(t) = α t , which is naturally linked to Nesterov's accelerated method.…”

Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators

“…Corresponding results for the algorithmic case have been obtained by Chambolle-Dossal [20] and Attouch-Peypouquet [8]. Independently, and mainly motivated by applications to partial differential equations and control problems, Jendoubi-May [22] and Attouch-Cabot [3,4] consider more general time-dependent damping coefficient γ(·). The latter includes the corresponding forward-backward algorithms, and unifies previous results.…”

Rate of convergence of the Nesterov accelerated gradient method in the subcritical case α ≤ 3

“…That is the situation we are now considering. For other cases of strong convergence (solution set with non-empty interior, even functions) one can consult [5], [2].…”

“…Under this condition, we are able to show the strong convergence of the trajectories to this unique minimum, and determine precisely the decay rate of the energy W along the trajectories. Indeed, an interesting property that has been put recentenly to the fore in [2], [5], [36] is that the convergence rates increase indefinitely with larger values of α for these functions. The following theorem completes these results by examinating also the case α ≤ 3 and gives a synthetic view of this situation.…”