FAST CONVEX OPTIMIZATION VIA A THIRD-ORDER IN TIME EVOLUTION EQUATION

In a Hilbert space H, we develop fast convex optimization methods, which are based on a third order in time evolution system. The function to minimize f : H → R is convex, continuously differentiable, with argmin f = ∅, and enters the dynamic via its gradient. On the basis of Lyapunov's analysis and temporal scaling techniques, we show a convergence rate of the values of the order 1/t 3 , and obtain the convergence of the trajectories towards optimal solutions. When f is strongly convex, an exponential rate of convergence is obtained. We complete the study of the continuous dynamic by introducing a damping term induced by the Hessian of f. This allows the oscillations to be controlled and attenuated. Then, we analyze the convergence of the proximal-based algorithms obtained by temporal discretization of this system, and obtain similar convergence rates. The algorithmic results are valid for a general convex, lower semicontinuous, and proper function f : H → R ∪ {+∞}