Convergence of inertial dynamics and proximal algorithms governed by maximally monotone operators

被引:0
|
作者
Hedy Attouch
Juan Peypouquet
机构
[1] Université Montpellier,Institut Montpelliérain Alexander Grothendieck, UMR 5149 CNRS
[2] Universidad Técnica Federico Santa María,Departamento de Matemática
[3] Universidad de Chile,Departamento de Ingeniería Matemática & Centro de Modelamiento Matemático (CNRS UMI2807), FCFM
来源
Mathematical Programming | 2019年 / 174卷
关键词
Asymptotic stabilization; Large step proximal method; Damped inertial dynamics; Lyapunov analysis; Maximally monotone operators; Time-dependent viscosity; Vanishing viscosity; Yosida regularization; 37N40; 46N10; 49M30; 65K05; 65K10; 90B50; 90C25;
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摘要
We study the behavior of the trajectories of a second-order differential equation with vanishing damping, governed by the Yosida regularization of a maximally monotone operator with time-varying index, along with a new Regularized Inertial Proximal Algorithm obtained by means of a convenient finite-difference discretization. These systems are the counterpart to accelerated forward–backward algorithms in the context of maximally monotone operators. A proper tuning of the parameters allows us to prove the weak convergence of the trajectories to zeroes of the operator. Moreover, it is possible to estimate the rate at which the speed and acceleration vanish. We also study the effect of perturbations or computational errors that leave the convergence properties unchanged. We also analyze a growth condition under which strong convergence can be guaranteed. A simple example shows the criticality of the assumptions on the Yosida approximation parameter, and allows us to illustrate the behavior of these systems compared with some of their close relatives.
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页码:391 / 432
页数:41
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