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, Volume 139, Issue 1, pp 115137
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Convergence of inexact Newton methods for generalized equations
 A. L. DontchevAffiliated withMathematical Reviews Email author
 , R. T. RockafellarAffiliated withDepartment of Mathematics, University of Washington
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For solving the generalized equation \(f(x)+F(x) \ni 0\), where \(f\) is a smooth function and \(F\) is a setvalued mapping acting between Banach spaces, we study the inexact Newton method described by where \(Df\) is the derivative of \(f\) and the sequence of mappings \(R_k\) represents the inexactness. We show how regularity properties of the mappings \(f+F\) and \(R_k\) are able to guarantee that every sequence generated by the method is convergent either qlinearly, qsuperlinearly, or qquadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.
$$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$
Keywords
Inexact Newton method Generalized equations Metric regularity Metric subregularity Variational inequality Nonlinear programmingMathematics Subject Classification (2000)
49J53 49K40 49M37 65J15 90C31 Title
 Convergence of inexact Newton methods for generalized equations
 Journal

Mathematical Programming
Volume 139, Issue 12 , pp 115137
 Cover Date
 201306
 DOI
 10.1007/s101070130664x
 Print ISSN
 00255610
 Online ISSN
 14364646
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Inexact Newton method
 Generalized equations
 Metric regularity
 Metric subregularity
 Variational inequality
 Nonlinear programming
 49J53
 49K40
 49M37
 65J15
 90C31
 Industry Sectors
 Authors

 A. L. Dontchev ^{(1)}
 R. T. Rockafellar ^{(2)}
 Author Affiliations

 1. Mathematical Reviews, Ann Arbor, MI, 481078604, USA
 2. Department of Mathematics, University of Washington, Seattle, WA, 981954350, USA