# An extension of the second order dynamical system that models Nesterov's convex gradient method

## TL;DR

This paper introduces an extended second order dynamical system related to Nesterov's method, demonstrating convergence properties and smoothing effects, bridging continuous dynamics and discrete inertial algorithms for convex optimization.

## Contribution

It extends existing dynamical systems to model Nesterov's algorithm, showing convergence and smoothing effects through theoretical analysis and numerical experiments.

## Key findings

- Objective function value converges at rate O(1/t^2)
- Trajectory converges to a minimum point
- Smoothing effect reduces oscillations in the energy error

## Abstract

In this paper we deal with a general second order continuous dynamical system associated to a convex minimization problem with a Fr\`echet differentiable objective function. We show that inertial algorithms, such as Nesterov's algorithm, can be obtained via the natural explicit discretization from our dynamical system. Our dynamical system can be viewed as a perturbed version of the heavy ball method with vanishing damping, however the perturbation is made in the argument of the gradient of the objective function. This perturbation seems to have a smoothing effect for the energy error and eliminates the oscillations obtained for this error in the case of the heavy ball method with vanishing damping, as some numerical experiments show. We prove that the value of the objective function in a generated trajectory converges in order O(1/t^2) to the global minimum of the objective function. Moreover, we obtain that a trajectory generated by the dynamical system converges to a minimum point of the objective function.