Abstract
In a Hilbert space H, in order to develop fast optimization methods, we analyze the asymptotic behavior, as time t tends to infinity, of inertial continuous dynamics where the damping acts as a closed-loop control. The function f : H → R to be minimized (not necessarily convex) enters the dynamic through it gradient, which is assumed to be Lipschitz continuous on the bounded subsets of H. This gives autonomous dynamical systems with nonlinear damping and nonlinear driving force. We first consider the case where the damping term ∂φ(ẋ(t)) acts as a closed-loop control of the velocity. The damping potential φ : H → R + is a convex continuous function which achieves its minimum at the origin. We show the existence and uniqueness of a global solution to the associated Cauchy problem. Then, we analyze the asymptotic convergence properties of the trajectories generated by this system. To do this, we use techniques from optimization, control theory, and PDE's: Lyapunov analysis based on the decreasing property of an energy-like function, quasi-gradient and Kurdyka-Lojasiewicz theory, monotone operator theory for wave-like equations. Convergence rates are obtained based on the geometric properties of the data f and φ. We put forward minimal hypotheses on the damping term guaranteeing the convergence of trajectories, thus showing the dividing line between strong and weak damping. When f is strongly convex, we give general conditions on the damping potential φ which provide exponential convergence rates. Then, we extend the results to the case where an additional Hessian-driven damping enters the dynamic, which reduces the oscillations. Finally, we consider an inertial system with a closed-loop damping involving jointly the velocityẋ(t) and the gradient ∇f (x(t)). This study naturally leads to similar results for the proximal-gradient algorithms which can be derived by temporal discretization, some of them are studied in the article. In addition to its original results, this work surveys the numerous works devoted in recent years to the interaction between continuous damped inertial dynamics and numerical algorithms for optimization, with the emphasis on autonomous systems, closed-loop adaptive procedures, and convergence rates.