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| Format: | Preprint |
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2015
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| Online Access: | https://arxiv.org/abs/1504.07793 |
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| _version_ | 1866913561160187904 |
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| author | Abbas, Boushra |
| author_facet | Abbas, Boushra |
| contents | Let $Φ:\mathcal{H}\longrightarrow\mathbb{R\cup}\left\{ +\infty\right\} $ be a closed convex proper function on a real Hilbert space $\mathcal{H}$, and $\partialΦ:\mathcal{H}\rightrightarrows\mathcal{H}$ its subdifferential. For any control function $ε:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ which tends to zero as $t$ goes to $+\infty$, and $λ$ a positive parameter, we study the asymptotic behavior of the trajectories of the regularized Newton dynamical system \begin{eqnarray*}
& & \upsilon\left(t\right)\in\partialΦ\left(x\left(t\right)\right)
& & λ\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 $C=\textrm{argmin}Φ\neq\emptyset$, 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 $Φ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1504_07793 |
| institution | arXiv |
| publishDate | 2015 |
| record_format | arxiv |
| spellingShingle | An asymptotic viscosity selection result for the regularized Newton dynamic Abbas, Boushra Optimization and Control Let $Φ:\mathcal{H}\longrightarrow\mathbb{R\cup}\left\{ +\infty\right\} $ be a closed convex proper function on a real Hilbert space $\mathcal{H}$, and $\partialΦ:\mathcal{H}\rightrightarrows\mathcal{H}$ its subdifferential. For any control function $ε:\mathbb{R}_{+}\longrightarrow\mathbb{R}_{+}$ which tends to zero as $t$ goes to $+\infty$, and $λ$ a positive parameter, we study the asymptotic behavior of the trajectories of the regularized Newton dynamical system \begin{eqnarray*} & & \upsilon\left(t\right)\in\partialΦ\left(x\left(t\right)\right) & & λ\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 $C=\textrm{argmin}Φ\neq\emptyset$, 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 $Φ$ 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. |
| title | An asymptotic viscosity selection result for the regularized Newton dynamic |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/1504.07793 |