# The long time behavior and the rate of convergence of symplectic convex algorithms obtained via splitting discretizations of inertial damping systems

## Abstract

In this paper we propose new numerical algorithms in the setting of unconstrained optimization problems and we give proof for the rate of convergence in the iterates of the objective function.

Furthermore, our algorithms are based upon splitting and symplectic methods and they preserve the energy properties of the inherent continuous dynamical system that contains a Hessian perturbation.

At the same time, we show that Nesterov gradient method is equivalent to a Lie-Trotter splitting applied to a Hessian driven damping system.

Finally, some numerical experiments are presented in order to validate the theoretical results.

## Authors

Cristian Daniel Alecsa
Tiberiu Popoviciu Institute of Numerical Analysis Romanian Academy Cluj-Napoca, Romania
Romanian Institute of Science and Technology, Cluj-Napoca, Romania

## Keywords

unconstrained optimization; rate of convergence; inertial algorithms; Nesterov; convex function; Hessian; dynamical system; splitting operator; vanishing damping; Lie-Trotter.

## Paper coordinates

Cristian-Daniel Alecsa, The long time behavior and the rate of convergence of symplectic convex algorithms obtained via splitting discretizations of inertial damping systems, arXiv:2001.10831