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An accelerated differential equation system for generalized equations

  • * Corresponding author: Qiyuan Wei

    * Corresponding author: Qiyuan Wei 

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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  • 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 $.

    Mathematics Subject Classification: 90C30.

    Citation:

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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
     | Show Table
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