An accelerated differential equation system for generalized equations

Qiyuan Wei; Liwei Zhang

doi:10.3934/jimo.2021195

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2023, Volume 19, Issue 1: 529-548. Doi: 10.3934/jimo.2021195

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An accelerated differential equation system for generalized equations

Qiyuan Wei^, and
Liwei Zhang

Institute of Operations Research and Control Theory, School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

^* Corresponding author: Qiyuan Wei
^* Corresponding author: Qiyuan Wei

Received: May 2021

Early access: November 2021

Published: January 2023

Supported by the National Natural Science Foundation of China under project No.11971089 and No.11731013. Partially supported by Dalian High-level Talent Innovation Project No. 2020RD09

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Abstract

An accelerated differential equation system with Yosida regularization and its numerical discretized scheme, for solving solutions to a generalized equation, are investigated. Given a maximal monotone operator $ T $ on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation

$ \begin{equation} \dot{x}(t)+T_{\lambda(t)}(x(t)-\alpha(t)T_{\lambda(t)}(x(t))) = 0,\quad t\geq t_0\geq 0, \end{equation} $

to the solution set $ T^{-1}(0) $ of a generalized equation $ 0 \in T(x) $. With smart choices of parameters $ \lambda(t) $ and $ \alpha(t) $, we prove the weak convergence of the trajectory to some point of $ T^{-1}(0) $ with $ \|\dot{x}(t)\|\leq {\rm O}(1/t) $ as $ t\rightarrow +\infty $. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory $ x^{h}(t) \rightarrow x(t) $ on interval $ [0,+\infty) $ is demonstrated when the step size $ h \rightarrow 0 $.

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Mathematics Subject Classification: 90C30.

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Table 1. Convergence of multipliers with different $ h $ step size

$ h $	$ \mu_1 $	$ \mu_2 $	$ \mu_3 $
0.5	0.4004	1.0000	1.1990
0.1	0.3714	1.0000	1.2570
0.05	0.3681	1.0000	1.2640
0.02	0.3661	1.0000	1.2580
0.01	0.3655	1.0000	1.2690
0.005	0.3651	1.0000	1.2700
0.001	0.3649	1.0000	1.2700

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