SOLVING QUADRATICALLY CONSTRAINED LEAST SQUARES PROBLEMS USING 
A DIFFERENTIAL-GEOMETRIC APPROACH

Lars Eld\'en
Department of Mathematics, Link\"oping
University
SE-581 83, Link\"oping, Sweden
email: laeld@math.liu.se

Report LiTH-MAT-R-2001-04

March 5, 2001


ABSTRACT

A quadratically constrained linear least squares problem is usually
solved using a Lagrange multiplier for the constraint and then solving
numerically 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 very slow. 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 its convergence is
often quite fast it appears to be superior to the Lagrange multiplier
method. A few numerical examples are given.