Generalized Newton’s method based on graphical derivatives

This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type metho...

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Bibliographic Details
Published in:Nonlinear analysis Vol. 75; no. 3; pp. 1324 - 1340
Main Authors: Hoheisel, T., Kanzow, C., Mordukhovich, B.S., Phan, H.
Format: Journal Article
Language:English
Published: Amsterdam Elsevier Ltd 01.02.2012
Elsevier
Subjects:
Algorithms
Calculus of variations and optimal control
Convergence
Derivatives
Differentiation
Exact sciences and technology
Global analysis, analysis on manifolds
Graphical derivatives and coderivatives
Local and global convergence
Mathematical analysis
Mathematics
Newton methods
Newton’s method
Nonlinear equations
Nonlinearity
Nonsmooth equations
Numerical analysis
Numerical analysis. Scientific computation
Numerical approximation
Numerical methods in mathematical programming, optimization and calculus of variations
Numerical methods in optimization and calculus of variations
Optimization and variational analysis
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Nonsmooth equations
65K15
Graphical derivatives and coderivatives
90C30
49J53
Local and global convergence
Newton’s method
Optimization and variational analysis
Nonlinear analysis
Optimization method
Newton's method
Mathematical model
Newton method
ISSN:0362-546X, 1873-5215
Online Access:Get full text
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Summary:This paper concerns developing a numerical method of the Newton type to solve systems of nonlinear equations described by nonsmooth continuous functions. We propose and justify a new generalized Newton algorithm based on graphical derivatives, which have never been used to derive a Newton-type method for solving nonsmooth equations. Based on advanced techniques of variational analysis and generalized differentiation, we establish the well-posedness of the algorithm, its local superlinear convergence, and its global convergence of the Kantorovich type. Our convergence results hold with no semismoothness and Lipschitzian assumptions, which is illustrated by examples. The algorithm and main results obtained in the paper are compared with well-recognized semismooth and B -differentiable versions of Newton’s method for nonsmooth Lipschitzian equations.
Bibliography:ObjectType-Article-2
SourceType-Scholarly Journals-1
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ISSN:0362-546X
1873-5215
DOI:10.1016/j.na.2011.06.039