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 → + ∞ of the trajectories of the second-order differential equation ddotx(t) + fracαt dotx(t) + nabla Phi (x(t)) = g(t), where nablaPhi is the gradient of a convex continuously differentiable function Phi: mathcal H → mathbb R, α is a positive parameter, and g: [t0, + ∞[ rightarrow mathcal H is a "small" perturbation term. In this damped inertial system, the viscous damping coefficient fracαt vanishes asymptotically, but not too rapidly. For α ≥ 3, and intt0+∞ t |g(t)| dt < + ∞, 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))- minmathcal HPhi ≤ fracCt2. For α > 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.