On Weighted Linear Least-Squares Problems Related to Interior Methods for Convex Quadratic Programming

Abstract

It is known that the norm of the solution to a weighted linear least-squares problem is uniformly bounded for the set of diagonally dominant symmetric positive definite weight matrices. This result is extended to weight matrices that are nonnegative linear combinations of symmetric positive semidefinite matrices. Further, results are given concerning the strong connection between the boundedness of weighted projection onto a subspace and the projection onto its complementary subspace using the inverse weight matrix. In particular, explicit bounds are given for the Euclidean norm of the projections. These results are applied to the Newton equations arising in a primal-dual interior method for convex quadratic programming and boundedness is shown for the corresponding projection operator.