Header

UZH-Logo

Maintenance Infos

Nonsmooth Kantorovich–Newton Methods: Hypotheses and Auxiliary Problems


Klatte, Diethard; Kummer, Bernd (2019). Nonsmooth Kantorovich–Newton Methods: Hypotheses and Auxiliary Problems. Vietnam Journal of Mathematics, 47(3):639-657.

Abstract

In Newton's method $0 \in f(x_k) + G(x_k) (x_{k+1} - x_k)$ for solving a nonsmooth equation $f(x) = 0$, the type of approximation of $f$ by some (generally multivalued) mapping $G$ determines not only the convergence behavior of the method, but also the difficulty and the concrete form of the auxiliary problems. With $G(x)=\partial f(x)$
(Clarke's Jacobian) -- like for locally convergent semismooth Newton methods -- and for various other generalized "derivatives", the inclusion is a canonical one, i.e., it describes the usual Newton step if $f$ is continuously differentiable near $x_k$. In our paper, we are interested in Kantorovich--type statements of convergence and study which meaningful hypotheses and auxiliary problems for particular pairs $(f,G)$ may replace those of the classical smooth case. In particular, we point out -- theoretically and by an example -- why the related hypotheses cannot be checked for canonical methods even if $f$ is piecewise linear.

Abstract

In Newton's method $0 \in f(x_k) + G(x_k) (x_{k+1} - x_k)$ for solving a nonsmooth equation $f(x) = 0$, the type of approximation of $f$ by some (generally multivalued) mapping $G$ determines not only the convergence behavior of the method, but also the difficulty and the concrete form of the auxiliary problems. With $G(x)=\partial f(x)$
(Clarke's Jacobian) -- like for locally convergent semismooth Newton methods -- and for various other generalized "derivatives", the inclusion is a canonical one, i.e., it describes the usual Newton step if $f$ is continuously differentiable near $x_k$. In our paper, we are interested in Kantorovich--type statements of convergence and study which meaningful hypotheses and auxiliary problems for particular pairs $(f,G)$ may replace those of the classical smooth case. In particular, we point out -- theoretically and by an example -- why the related hypotheses cannot be checked for canonical methods even if $f$ is piecewise linear.

Statistics

Citations

Altmetrics

Downloads

1 download since deposited on 05 Feb 2020
1 download since 12 months
Detailed statistics

Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:03 Faculty of Economics > Department of Business Administration
Dewey Decimal Classification:330 Economics
Scopus Subject Areas:Physical Sciences > General Mathematics
Uncontrolled Keywords:General Mathematics
Language:English
Date:1 September 2019
Deposited On:05 Feb 2020 07:20
Last Modified:29 Jul 2020 13:37
Publisher:Springer
ISSN:2305-221X
OA Status:Closed
Publisher DOI:https://doi.org/10.1007/s10013-019-00348-4

Download

Closed Access: Download allowed only for UZH members

Content: Accepted Version
Language: English
Filetype: PDF - Registered users only until 30 September 2020
Size: 312kB
View at publisher
Embargo till: 2020-09-30