In this paper we deal with a general second order continuous dynamical system associated to a convex minimization problem with a Fréchet 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 \(\mathcal {O}(1/t^2)\) 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.

在本文中，我们处理与具有Fréchet可微目标函数的凸极小化问题相关的一般二阶连续动力系统。我们证明了惯性算法，例如Nesterov算法，可以通过我们的动力学系统通过自然的显式离散化来获得。我们的动力学系统可以看作是阻尼消失的重球法的一种扰动形式，但是这种扰动是由目标函数的梯度决定的。如一些数值实验所示，这种扰动似乎对能量误差具有平滑效果，并且消除了在阻尼消失的重球法情况下针对该误差获得的振荡。我们证明目标函数在生成的轨迹中的值按顺序收敛\（\ mathcal {O}（1 / t ^ 2）\）到目标函数的全局最小值。此外，我们获得了由动力系统生成的轨迹收敛到目标函数的最小点。