On Linear Least-Squares Problems with Diagonally Dominant Weight Matrices

The solution of the unconstrained weighted linear least-squares problem is known to be a convex combination of the basic solutions formed by the nonsingular subsystems if the weight matrix is diagonal andpositive definite, which implies that the norm of this solution is uniformly bounded for any diagonal and positive definite weight matrix.

Abstract

The solution of the unconstrained weighted linear least-squares problem is known to be a convex combination of the basic solutions formed by the nonsingular subsystems if the weight matrix is diagonal and positive definite. In particular, this implies that the norm of this solution is uniformly bounded for any diagonal and positive definite weight matrix. In addition, the solution set is known to be the relative interior of a finite set of polytopes if the weight matrix varies over the set of positive definite diagonal matrices. In this paper, these results are reviewed and generalized to the set of weight matrices that are symmetric, positive semidefinite, and diagonally dominant and that give unique solution to the least-squares problem. This is done by means of a particular symmetric diagonal decomposition of the weight matrix, giving a finite number of diagonally weighted problems but in a space of higher dimension. Extensions to equality-constrained weighted linear least-squares problems are given. A discussion of why the boundedness properties do not hold for general symmetric positive definite weight matrices is given. The motivation for this research is from interior methods for optimization.