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Solving quadratically constrained least squares problems using a differential-geometric approach
Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Scientific Computing.ORCID iD: 0000-0003-2281-856X
2002 (English)In: BIT Numerical Mathematics, ISSN 0006-3835, Vol. 42, no 2, 323-335 p.Article in journal (Refereed) Published
Abstract [en]

A quadratically constrained linea least squares problem is usually solved using a Lagrange multiplier for the constraint and then solving iteratively a nonlinear secular equation for the optimal Lagrange multiplier. It is well-known that, due to the closeness to a pole for the secular equation, standard methods for solving the secular equation can be slow, and sometimes it is not easy to select a good starting value for the iteration. The problem can be reformulated as that of minimizing the residual of the least squares problem on the unit sphere. Using a differential-geometric approach we formulate Newton's method on the sphere, and thereby avoid the difficulties associated with the Lagrange multiplier formulation. This Newton method on the sphere can be implemented efficiently, and since it is easy to find a good starting value for the iteration, and the convergence is often quite fast, it has a clear advantage over the Lagrange multiplier method. A numerical example is given.

Place, publisher, year, edition, pages
2002. Vol. 42, no 2, 323-335 p.
Keyword [en]
ill-conditioned, Lagrange multiplier, least squares, Newton's method, quadratic constraint, Stiefel manifold
National Category
Engineering and Technology
URN: urn:nbn:se:liu:diva-48903OAI: diva2:269799
Available from: 2009-10-11 Created: 2009-10-11 Last updated: 2013-08-30

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