Abstract
In a Hilbert space setting H, for convex optimization, we analyze the fast convergence properties as t → +∞ of the trajectories t → u(t) ∈ H generated by a third-order in time evolution system. The function f : H → R to minimize is supposed to be convex, continuously differentiable, with argmin H f = ∅. It enters into the dynamic through its gradient. Based on this new dynamical system, we improve the results obtained by [Attouch, Chbani, Riahi: Fast convex optimization via a third-order in time evolution equation, Optimization 2020]. As a main result, when the damping parameter α satisfies α > 3, we show that f (u(t)) − inf H f = o 1/t 3 as t → +∞, as well as the convergence of the trajectories. We complement these results by introducing into the dynamic an Hessian driven damping term, which reduces the oscillations. In the case of a strongly convex function f , we show an autonomous evolution system of the third order in time with an exponential rate of convergence. All these results have natural extensions to the case of a convex lower semicontinuous function f : H → R ∪ {+∞}. Just replace f with its Moreau envelope.