Generalized equations: Solvability and regularity

1984 ◽  
pp. 199-212 ◽  
Author(s):  
Bernd Kummer
Keyword(s):  
Generalized Equations

A Method of Including the Effects of Main Flux Path Saturation in the Generalized Equations of A.C. Machines

1983 ◽  
Vol PER-3 (1) ◽  
pp. 33-33
Author(s):  
J. E. Brown ◽  
K. P. Kovacs ◽  
P. Vas
Keyword(s):  
Generalized Equations ◽  
Flux Path

Newton's method for fully parameterized generalized equations

Optimization ◽  
2018 ◽  
Vol 67 (11) ◽  
pp. 2061-2080 ◽  
Author(s):  
Wei Ouyang ◽  
Binbin Zhang
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Generalized Equations

Generalized equations and Newton’s and method

2017 ◽  
pp. 193-203
Author(s):  
Ioannis K. Argyros ◽  
Á. Alberto Magreñán
Keyword(s):  
Generalized Equations

Newton’s method with feasible inexact projections for solving constrained generalized equations

2018 ◽  
Vol 72 (1) ◽  
pp. 159-177 ◽  
Author(s):  
Fabiana R. de Oliveira ◽  
Orizon P. Ferreira ◽  
Gilson N. Silva
Keyword(s):  
Newton’S Method ◽  
Newton's Method ◽  
Generalized Equations

scholarly journals An accelerated differential equation system for generalized equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Qiyuan Wei ◽  
Liwei Zhang
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Maximal Monotone ◽  
Equation System ◽  
Generalized Equation ◽  
Generalized Equations ◽  
Differential Equation System ◽  
Step Size ◽  
Yosida Regularization ◽  
Numerical Discretization

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ T $\end{document}</tex-math></inline-formula> on a Hilbert space, this paper will study the asymptotic behavior of the solution trajectories of the differential equation</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>to the solution set <inline-formula><tex-math id="M2">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> of a generalized equation <inline-formula><tex-math id="M3">\begin{document}$ 0 \in T(x) $\end{document}</tex-math></inline-formula>. With smart choices of parameters <inline-formula><tex-math id="M4">\begin{document}$ \lambda(t) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \alpha(t) $\end{document}</tex-math></inline-formula>, we prove the weak convergence of the trajectory to some point of <inline-formula><tex-math id="M6">\begin{document}$ T^{-1}(0) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M7">\begin{document}$ \|\dot{x}(t)\|\leq {\rm O}(1/t) $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M8">\begin{document}$ t\rightarrow +\infty $\end{document}</tex-math></inline-formula>. Interestingly, under the upper Lipshitzian condition, strong convergence and faster convergence can be obtained. For numerical discretization of the system, the uniform convergence of the Euler approximate trajectory <inline-formula><tex-math id="M9">\begin{document}$ x^{h}(t) \rightarrow x(t) $\end{document}</tex-math></inline-formula> on interval <inline-formula><tex-math id="M10">\begin{document}$ [0,+\infty) $\end{document}</tex-math></inline-formula> is demonstrated when the step size <inline-formula><tex-math id="M11">\begin{document}$ h \rightarrow 0 $\end{document}</tex-math></inline-formula>.</p>

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