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In a Hilbert space, we develop fast convex optimization methods which are based on an evolution system of the third-order in time. The function to minimizeis convex, continuously differentiable, with, 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, 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 analyse 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.

Fast convex optimization via a third-order in time evolution equation