Not logged in.

# Contribution Details

 Type Journal Article Scope Discipline-based scholarship Title Approximations and generalized Newton methods Organization Unit Authors Diethard Klatte Bernd Kummer Item Subtype Original Work Refereed Yes Status Published electronically before print/final form (Epub ahead of print) Language English Journal Title Mathematical Programming: Series B Publisher Springer Geographical Reach international ISSN 0025-5610 Page Range Epub ahead of print Date 2017 Abstract Text 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, 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, 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. Free access at DOI Digital Object Identifier 10.1007/s10107-017-1194-8 PDF File Download from ZORA Export BibTeX EP3 XML (ZORA) Keywords Generalized Newton method, local convergence, inclusion, generalized equation, regularity, Newton map, nonlinear approximation, successive approximation