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Volume 40, Issue 1
Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method

Mohammed Harunor Rashid

DOI: 10.4208/jcm.2005-m2019-0019

J. Comp. Math., 40 (2022), pp. 44-69.

Published online: 2021-11

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  • Abstract

In the present paper, we study the restricted inexact Newton-type method for solving the generalized equation $0\in f(x)+F(x)$, where $X$ and $Y$ are Banach spaces, $f:X\to Y$ is a Fréchet differentiable function and $F\colon X\rightrightarrows Y$ is a set-valued mapping with closed graph. We establish the convergence criteria of the restricted inexact Newton-type method, which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Fréchet derivative of $f$. Indeed, we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Fréchet derivative of $f$ is continuous and Lipschitz continuous as well as $f+F$ is metrically regular. An application of this method to variational inequality is given. In addition, a numerical experiment is given which illustrates the theoretical result.

  • Keywords

Generalized equation, Restricted inexact Newton-type method, Metrically regular mapping, Partial Lipschitz-like mapping, Semilocal convergence.

  • AMS Subject Headings

47H04, 49J53, 65K10, 90C30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

harun_math@ru.ac.bd (Mohammed Harunor Rashid)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-44, author = {Harunor Rashid , Mohammed}, title = {Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method}, journal = {Journal of Computational Mathematics}, year = {2021}, volume = {40}, number = {1}, pages = {44--69}, abstract = {

In the present paper, we study the restricted inexact Newton-type method for solving the generalized equation $0\in f(x)+F(x)$, where $X$ and $Y$ are Banach spaces, $f:X\to Y$ is a Fréchet differentiable function and $F\colon X\rightrightarrows Y$ is a set-valued mapping with closed graph. We establish the convergence criteria of the restricted inexact Newton-type method, which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Fréchet derivative of $f$. Indeed, we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Fréchet derivative of $f$ is continuous and Lipschitz continuous as well as $f+F$ is metrically regular. An application of this method to variational inequality is given. In addition, a numerical experiment is given which illustrates the theoretical result.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2005-m2019-0019}, url = {http://global-sci.org/intro/article_detail/jcm/19969.html} }
TY - JOUR T1 - Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method AU - Harunor Rashid , Mohammed JO - Journal of Computational Mathematics VL - 1 SP - 44 EP - 69 PY - 2021 DA - 2021/11 SN - 40 DO - http://doi.org/10.4208/jcm.2005-m2019-0019 UR - https://global-sci.org/intro/article_detail/jcm/19969.html KW - Generalized equation, Restricted inexact Newton-type method, Metrically regular mapping, Partial Lipschitz-like mapping, Semilocal convergence. AB -

In the present paper, we study the restricted inexact Newton-type method for solving the generalized equation $0\in f(x)+F(x)$, where $X$ and $Y$ are Banach spaces, $f:X\to Y$ is a Fréchet differentiable function and $F\colon X\rightrightarrows Y$ is a set-valued mapping with closed graph. We establish the convergence criteria of the restricted inexact Newton-type method, which guarantees the existence of any sequence generated by this method and show this generated sequence is convergent linearly and quadratically according to the particular assumptions on the Fréchet derivative of $f$. Indeed, we obtain semilocal and local convergence results of restricted inexact Newton-type method for solving the above generalized equation when the Fréchet derivative of $f$ is continuous and Lipschitz continuous as well as $f+F$ is metrically regular. An application of this method to variational inequality is given. In addition, a numerical experiment is given which illustrates the theoretical result.

Harunor Rashid , Mohammed. (2021). Metrically Regular Mapping and Its Utilization to Convergence Analysis of a Restricted Inexact Newton-Type Method. Journal of Computational Mathematics. 40 (1). 44-69. doi:10.4208/jcm.2005-m2019-0019
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