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Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
2004 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 73, no 248, p. 1865-1883Article in journal (Refereed) Published
##### Abstract [en]

A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as \$min_{x} 1/2Vert f_{2}(x) Vert _{2}^{2}\$subject to the constraints \$f_{1}(x) = 0\$, the Jacobian \$J_{1} = partial f_{1}/ partial x\$ and/or the Jacobian \$J = partial f/ partial x\$, \$f = [f_{1};f_{2}]\$, may be ill conditioned at the solution. We analyze the important special case when \$J_{1}\$ and/or \$J\$ do not have full rank at the solution. In order to solve such a problem, we formulate a nonlinear least norm problem. Next we describe a truncated Gauss-Newton method. We show that the local convergence rate is determined by the maximum of three independent Rayleigh quotients related to three different spaces in \$mathbb{R} ^{n}\$. Another way of solving an ill-posed nonlinear least squares problem is to regularize the problem with some parameter that is reduced as the iterates converge to the minimum. Our approach is a Tikhonov based local linear regularization that converges to a minimum norm problem. This approach may be used both for almost and rank-deficient Jacobians. Finally we present computational tests on constructed problems verifying the local analysis.

##### Place, publisher, year, edition, pages
2004. Vol. 73, no 248, p. 1865-1883
##### Keywords [en]
Nonlinear least squares, nonlinear constraints, optimization, regularization, Gauss-Newton method
##### National Category
Computer Sciences
##### Identifiers
ISI: 000222002400014Scopus ID: 2-s2.0-5744234026Local ID: 4383OAI: oai:DiVA.org:miun-3968DiVA, id: diva2:29000
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2018-01-12Bibliographically approved

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Gulliksson, Mårten E.

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CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
• apa
• ieee
• modern-language-association-8th-edition
• vancouver
• Other style
More styles
Language
• de-DE
• en-GB
• en-US
• fi-FI
• nn-NO
• nn-NB
• sv-SE
• Other locale
More languages
Output format
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• text
• asciidoc
• rtf