Fast inertial dynamics and FISTA algorithms in convex optimization. Perturbation aspects

In a Hilbert space setting $\mathcal H$, we study the fast convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation $\ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) = g(t)$, where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, $\alpha$ is a positive parameter, and $g: [t_0, + \infty[ \rightarrow \mathcal H$ is a "small" perturbation term. In this damped inertial system, the viscous damping coefficient $\frac{\alpha}{t}$ vanishes asymptotically, but not too rapidly. For $\alpha \geq 3$, and $\int_{t_0}^{+\infty} t \|g(t)\| dt < + \infty$, just assuming that the solution set is non empty, we show that any trajectory of the above system satisfies the fast convergence property $\Phi(x(t))- \min_{\mathcal H}\Phi \leq \frac{C}{t^2}$. For $\alpha > 3$, we show that any trajectory converges weakly to a minimizer of $\Phi$, and we show the strong convergence property in various practical situations. This complements the results obtained by Su-Boyd- Cand\`es, and Attouch-Peypouquet-Redont, in the unperturbed case $g=0$. The parallel study of the time discretized version of this system provides new insight on the effect of errors, or perturbations on Nesterov's type algorithms. We obtain fast convergence of the values, and convergence of the trajectories for a perturbed version of the variant of FISTA recently considered by Chambolle-Dossal, and Su-Boyd-Cand\`es.