# An Iterative Procedure for Solving Nonsmooth Generalized Equation

• Volume: 34, Issue: 2, page 441-454
• ISSN: 1310-6600

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## Abstract

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2000 Mathematics Subject Classification: 47H04, 65K10.In this article, we study a general iterative procedure of the following form 0 ∈ f(xk)+F(xk+1), where f is a function and F is a set valued map acting from a Banach space X to a linear normed space Y, for solving generalized equations in the nonsmooth framework. We prove that this method is locally Q-linearly convergent to x* a solution of the generalized equation 0 ∈ f(x)+F(x) if the set-valued map [f(x*)+g(·)−g(x*)+F(·)]−1 is Aubin continuous at (0,x*), where g:X→ Y is a function, whose Fréchet derivative is L-Lipschitz.

## How to cite

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Marinov, Rumen Tsanev. "An Iterative Procedure for Solving Nonsmooth Generalized Equation." Serdica Mathematical Journal 34.2 (2008): 441-454. <http://eudml.org/doc/281452>.

@article{Marinov2008,
abstract = {2000 Mathematics Subject Classification: 47H04, 65K10.In this article, we study a general iterative procedure of the following form 0 ∈ f(xk)+F(xk+1), where f is a function and F is a set valued map acting from a Banach space X to a linear normed space Y, for solving generalized equations in the nonsmooth framework. We prove that this method is locally Q-linearly convergent to x* a solution of the generalized equation 0 ∈ f(x)+F(x) if the set-valued map [f(x*)+g(·)−g(x*)+F(·)]−1 is Aubin continuous at (0,x*), where g:X→ Y is a function, whose Fréchet derivative is L-Lipschitz.},
author = {Marinov, Rumen Tsanev},
journal = {Serdica Mathematical Journal},
keywords = {Set-Valued Maps; Generalized Equation; Linear Convergence; Aubin Continuity; set-valued maps; generalized equation; linear convergence; Aubin continuity; Banach space; normed space},
language = {eng},
number = {2},
pages = {441-454},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {An Iterative Procedure for Solving Nonsmooth Generalized Equation},
url = {http://eudml.org/doc/281452},
volume = {34},
year = {2008},
}

TY - JOUR
AU - Marinov, Rumen Tsanev
TI - An Iterative Procedure for Solving Nonsmooth Generalized Equation
JO - Serdica Mathematical Journal
PY - 2008
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 34
IS - 2
SP - 441
EP - 454
AB - 2000 Mathematics Subject Classification: 47H04, 65K10.In this article, we study a general iterative procedure of the following form 0 ∈ f(xk)+F(xk+1), where f is a function and F is a set valued map acting from a Banach space X to a linear normed space Y, for solving generalized equations in the nonsmooth framework. We prove that this method is locally Q-linearly convergent to x* a solution of the generalized equation 0 ∈ f(x)+F(x) if the set-valued map [f(x*)+g(·)−g(x*)+F(·)]−1 is Aubin continuous at (0,x*), where g:X→ Y is a function, whose Fréchet derivative is L-Lipschitz.
LA - eng
KW - Set-Valued Maps; Generalized Equation; Linear Convergence; Aubin Continuity; set-valued maps; generalized equation; linear convergence; Aubin continuity; Banach space; normed space
UR - http://eudml.org/doc/281452
ER -

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