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KKT conditions for rank-deficient nonlinear least-square problems with rank-deficient nonlinear constraints
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
1999 (English)In: Journal of Optimization Theory and Applications, Vol. 100, no 1, 145-160 p.Article in journal (Refereed) Published
Abstract [en]

In nonlinear least-square problems with nonlinear constraints, the function (1/2) // f(2)(x) // (2)(2), where f(2) is a nonlinear vector function, is to be minimized subject to the nonlinear constraints fi (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.
Identifiers
URN: urn:nbn:se:umu:diva-21941ISBN: 0022-3239 OAI: oai:DiVA.org:umu-21941DiVA: diva2:212198
Available from: 2009-04-21 Created: 2009-04-21 Last updated: 2009-04-21

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