Fast optimization via inertial dynamics with closed-loop damping

Abstract

In a real Hilbert space $\mathcal{H}$, in order to develop fast optimization methods, we analyze the asymptotic behavior, as time $t$ tends to infinity, of a large class of autonomous dissipative inertial continuous dynamics. The function $f:\mathcal{H}\to \mathbb{R}$ to be minimized (not necessarily convex) enters the dynamic via its gradient, which is assumed to be Lipschitz continuous on bounded subsets of $\mathcal{H}$. This results in autonomous dynamical systems with nonlinear damping and nonlinear driving force. We first consider the case where the damping term $\partial \varphi (\dot{x}(t))$ acts as closed-loop control of the velocity. The damping potential $\varphi :\mathcal{H}\to {\mathbb{R}}_{+}$ is a convex continuous function which reaches its minimum at the origin. We show the existence and uniqueness of a global solution to the associated Cauchy problem. We analyze the asymptotic convergence and the convergence rates of the trajectories generated by this system. To do this, we use techniques from optimization, control theory, and PDEs: Lyapunov analysis based on the decreasing property of an energy-like function, quasi-gradient and Kurdyka–Łojasiewicz theory, and monotone operator theory for wave-like equations. Convergence rates are obtained based on the geometric properties of the data $f$ and $\varphi$. We put forward minimal hypotheses on the damping potential $\varphi$ 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 $\varphi$ which provide exponential convergence rates. Then, we extend the results to the case where additional Hessian-driven damping enters the dynamic, which reduces the oscillations. Finally, we consider a new inertial system where damping jointly involves the velocity $\dot{x}(t)$ and the gradient $\nabla f(x(t))$. This study naturally leads to similar results for proximal-gradient algorithms obtained by temporal discretization; some of them are studied in the article. In addition to its original results, this work surveys numerous works devoted to the interaction between damped inertial continuous dynamics and numerical optimization algorithms, with an emphasis on autonomous systems, adaptive procedures, and convergence rates.