Abstract
In a Hilbert space setting , for convex optimization, we analyse the fast convergence properties as
of the trajectories
generated by a third-order in time evolution system. The function
to minimize is supposed to be convex, continuously differentiable, with
. 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
, we show that
as
, 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
. Just replace f with its Moreau envelope.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Notes
Throughout the paper,
is a real Hilbert space, endowed with the scalar product
and the associated norm
. Unless specified,
is a
convex function with
. We take
as the origin of time (this is justified by the singularity at the origin of the damping coefficient
which is used in the paper).