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Mathematical Programming

, Volume 168, Issue 1–2, pp 673–716 | Cite as

Approximations and generalized Newton methods

  • Diethard KlatteEmail author
  • Bernd Kummer
Full Length Paper Series B

Abstract

We present approaches to (generalized) Newton methods in the framework of generalized equations \(0\in f(x)+M(x)\), where f is a function and M is a multifunction. The Newton steps are defined by approximations \({\hat{f}}\) of f and the solutions of \(0\in {\hat{f}}(x)+M(x)\). We give a unified view of the local convergence analysis of such methods by connecting a certain type of approximation with the desired kind of convergence and different regularity conditions for \(f+M\). Our paper is, on the one hand, thought as a survey of crucial parts of the topic, where we mainly use concepts and results of the monograph (Klatte and Kummer in Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002). On the other hand, we present original results and new features. They concern the extension of convergence results via Newton maps (Klatte and Kummer in Nonsmooth equations in optimization: regularity, calculus, methods and applications, Kluwer Academic Publishers, Dordrecht, 2002; Kummer, in: Oettli, Pallaschke (eds) Advances in optimization, Springer, Berlin, 1992) from equations to generalized equations both for linear and nonlinear approximations \({\hat{f}}\), and relations between semi-smoothness, Newton maps and directional differentiability of f. We give a Kantorovich-type statement, valid for all sequences of Newton iterates under metric regularity, and recall and extend results on multivalued approximations for general inclusions \(0\in F(x)\). Equations with continuous, non-Lipschitzian f are considered, too.

Keywords

Generalized Newton method Local convergence Inclusion Generalized equation Regularity Newton map Nonlinear approximation Successive approximation 

Mathematics Subject Classification

49J53 49K40 90C31 65J05 

Notes

Acknowledgements

The authors are indebted to the referees for their constructive comments that significantly improved the presentation.

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Copyright information

© Springer-Verlag GmbH Germany and Mathematical Optimization Society 2017

Authors and Affiliations

  1. 1.IBWUniversität ZürichZurichSwitzerland
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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