Abstract
In a Hilbert space setting, in order to develop fast first-order methods for convex optimization, we study the asymptotic convergence properties (t → +∞) of the trajectories of the inertial dynamics \ddot{x}(t) + γ(t) \dot{x}(t) + β(t)\nabla Φ(x(t)) = 0. The function Φ to minimize is supposed to be convex, continuously differentiable, γ(t) is a positive damping coefficient, and β(t) is a time scale coefficient. Convergence rates for the values Φ(x(t)) − min Φ and the velocities are obtained under conditions involving only β(t) and γ(t). In this general framework (Φ is only assumed to be convex with a non-empty solution set), the fast convergence property is closely related to the asymptotic vanishing property γ(t) → 0, and to the temporal scaling β(t) → +∞. We show the optimality of the convergence rates thus obtained, and study their stability under external perturbation of the system. The discrete time versions of the results provide convergence rates for a large class of inertial proximal algorithms. In doing so, we improve the classical results on the subject, including Güler's inertial proximal algorithm.