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Approximations and generalized Newton methods


Klatte, Diethard; Kummer, Bernd (2018). Approximations and generalized Newton methods. Mathematical Programming: Series B, 168(1-2):673-716.

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, 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.

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, 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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Business Administration
Dewey Decimal Classification:330 Economics
510 Mathematics
Uncontrolled Keywords:Generalized Newton method, local convergence, inclusion, generalized equation, regularity, Newton map, nonlinear approximation, successive approximation
Language:English
Date:2018
Deposited On:19 Oct 2017 09:46
Last Modified:25 Feb 2018 02:01
Publisher:Springer
ISSN:0025-5610
OA Status:Closed
Free access at:Publisher DOI. An embargo period may apply.
Publisher DOI:https://doi.org/10.1007/s10107-017-1194-8

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Language: English
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Embargo till: 2018-09-11