# Convergence of damped inertial dynamics governed by regularized maximally monotone operators

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Abstract : In a Hilbert space setting, we study the asymptotic behavior, as time $t$ goes to infinity, of the trajectories of a second-order differential equation governed by the Yosida regularization of a maximally monotone operator with time-varying positive index $\lambda(t)$. The dissipative and convergence properties are attached to the presence of a viscous damping term with positive coefficient $\gamma(t)$. A suitable tuning of the parameters $\gamma(t)$ and $\lambda(t)$ makes it possible to prove the weak convergence of the trajectories towards zeros of the operator. When the operator is the subdifferential of a closed convex proper function, we estimate the rate of convergence of the values. These results are in line with the recent articles by Attouch-Cabot, and Attouch-Peypouquet. In this last paper, the authors considered the case $\gamma (t) =\frac{\alpha}{t}$, which is naturally linked to Nesterov's accelerated method. We unify, and often improve the results already present in the literature.
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Journal of Differential Equations, Elsevier, 2018, 264 (12), pp.7138-7182. 〈10.1016/j.jde.2018.02.017〉
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https://hal.archives-ouvertes.fr/hal-01648383
Contributeur : Alexandre Cabot <>
Soumis le : samedi 7 avril 2018 - 13:19:03
Dernière modification le : vendredi 8 juin 2018 - 14:50:07

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RIMS, final, JDE, Feb.19, 2018...
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Hedy Attouch, Alexandre Cabot. Convergence of damped inertial dynamics governed by regularized maximally monotone operators . Journal of Differential Equations, Elsevier, 2018, 264 (12), pp.7138-7182. 〈10.1016/j.jde.2018.02.017〉. 〈hal-01648383v2〉

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