An asymptotic viscosity selection result for the regularized Newton dynamic

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.