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KKT conditions for rank-deficient nonlinear least-square problems with rank-deficient nonlinear constraints
Responsible organisation
1999 (English)In: Medietrender och papperskvalitet, ISSN 0022-3239, Vol. 100, no 1, 145-160 p.Article in journal (Refereed) Published
Abstract [en]

In nonlinear least-square problems with nonlinear constraints, the function $$left. {(1/2)} right|left. {f_2 (x)} right|_2^2$$ , where f 2 is a nonlinear vector function, is to be minimized subject to the nonlinear constraints f 1(x)=0. This problem is ill-posed if the first-order KKT conditions do not define a locally unique solution. We show that the problem is ill-posed if either the Jacobian of f 1 or the Jacobian of J is rank-deficient (i.e., not of full rank) in a neighborhood of a solution satisfying the first-order KKT conditions. Either of these ill-posed cases makes it impossible to use a standard Gauss�Newton method. Therefore, we formulate a constrained least-norm problem that can be used when either of these ill-posed cases occur. By using the constant-rank theorem, we derive the necessary and sufficient conditions for a local minimum of this minimum-norm problem. The results given here are crucial for deriving methods solving the rank-deficient problem.

Place, publisher, year, edition, pages
1999. Vol. 100, no 1, 145-160 p.
Keyword [en]
Nonlinear least squares - optimization - regularization - KKT conditions - rank-deficient nonlinear constraints - rank-deficient nonlinear least-square problems
National Category
Mathematics
Identifiers
URN: urn:nbn:se:miun:diva-3805DOI: 10.1023/A:1021721132282Local ID: 4395OAI: oai:DiVA.org:miun-3805DiVA: diva2:28837
Available from: 2008-09-30 Created: 2009-09-21Bibliographically approved

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Gulliksson, Mårten
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CiteExportLink to record
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Cite
Citation style
  • apa
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  • de-DE
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  • nn-NB
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  • Other locale
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  • asciidoc
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