Fast convex optimization via a third-order in time evolution equation: TOGES-V an improved version of TOGES*

In a Hilbert space setting H, for convex optimization, we analyse 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 argminHf≠∅. It enters into the dynamic through its gradient. Based on this new dynamical system, we improve the results obtained by Attouch et al. [Fast convex optimization via a third-order in time evolution equation. Optimization. 2020;71(5):1275–1304]. As a main result, when the damping parameter α satisfies α>3, we show that f(u(t))−infHf=o(1/t3) 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.