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doi: 10.3934/jimo.2021195
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## An accelerated differential equation system for generalized equations

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

* Corresponding author: Qiyuan Wei

Received  May 2021 Early access November 2021

Fund Project: 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

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
 $$$\dot{x}(t)+T_{\lambda(t)}(x(t)-\alpha(t)T_{\lambda(t)}(x(t))) = 0,\quad t\geq t_0\geq 0,$$$
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
 $h \rightarrow 0$
.
Citation: Qiyuan Wei, Liwei Zhang. An accelerated differential equation system for generalized equations. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021195
##### References:

show all references

##### References:
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
 $h$ $\mu_1$ $\mu_2$ $\mu_3$ 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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