An asymptotic viscosity selection result for the regularized Newton dynamic

TL;DR

This paper investigates the long-term behavior of trajectories in a regularized Newton dynamic system within a Hilbert space, showing convergence to minimal norm solutions under certain conditions and analyzing the effects of different regularizations.

Contribution

It provides a detailed asymptotic analysis of the regularized Newton dynamic, demonstrating convergence to minimal norm solutions with a Tikhonov-like regularization in a convex setting.

Findings

Trajectories converge weakly to the minimal norm element of the solution set.

Strong convergence occurs when the function is differentiable with Lipschitz continuous gradient.

The regularization term acts as a Tikhonov regularization under moderate decay conditions.

Abstract

Let $\Phi:\mathcal{H}\longrightarrow\mathbb{R\cup}\left\{ +\infty\right\} $ be a closed convex proper function on a real Hilbert space $\mathcal{H}$, and $\partial\Phi:\mathcal{H}\rightrightarrows\mathcal{H}$ its subdifferential. For any control function $\epsilon:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ which tends to zero as $t$ goes to $+\infty$, and $\lambda$ a positive parameter, we study the asymptotic behavior of the trajectories of the regularized Newton dynamical system \begin{eqnarray*} & & \upsilon\left(t\right)\in\partial\Phi\left(x\left(t\right)\right) & & \lambda\dot{x}\left(t\right)+\dot{\upsilon}\left(t\right)+\upsilon\left(t\right)+\varepsilon\left(t\right)x\left(t\right)=0. \end{eqnarray*} Assuming that $\varepsilon\left(t\right)$ tends to zero moderately as $t$ goes to $+\infty$, we show that the term $\varepsilon\left(\cdot\right)x\left(\cdot\right)$ asymptotically acts as a Tikhonov regularization, which forces the trajectories to converge to a particular equilibrium. Precisely, when $C=\textrm{argmin}\Phi\neq\emptyset$, and $\varepsilon (\cdot)$ is a ``slow'' control, i.e., $\int_{0}^{+\infty}\varepsilon\left(t\right)dt=+\infty$, then each trajectory of the system converges weakly, as $t$ goes to $+\infty$, to the element of minimal norm of the closed convex set $C.$ When $\Phi$ is a convex differentiable function whose gradient is Lipschitz continuous, we show that the strong convergence property is satisfied. Then we examine the effect of other types of regularizing methods.