- AN ASYMPTOTIC VISCOSITY SELECTION RESULT FOR THE REGULARIZED NEWTON DYNAMIC Boushra Abbas(abbas.boushrahotmail.fr) 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. Keywords: Category 1: Nonlinear Optimization Citation: Download: [PDF]Entry Submitted: 04/29/2015Entry Accepted: 04/29/2015Entry Last Modified: 04/29/2015Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.