Equivalent formulae for the supremum and stability of weighted pseudoinverses

Abstract

During recent decades, there have been a great number of research articles studying interior-point methods for solving problems in mathematical programming and constrained optimization. Stewart and O’Leary obtained an upper bound for scaled pseudoinverses sup W ∈ P ‖ ( W 1 2 X ) + W 1 2 ‖ 2 \underset {W\in \mathcal {P} }{\text {sup}}\|(W^{\frac {1}{2}}X)^{+}W^{\frac {1}{2}}\|_{2} of a matrix X X where P \mathcal {P} is a set of diagonal positive definite matrices. We improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses. Forsgren further generalized these results to derive the supremum of weighted pseudoinverses sup W ∈ P ‖ ( W 1 2 X ) + W 1 2 ‖ 2 \underset {W\in \mathcal {P} }{\text {sup}}\|(W^{\frac {1}{2}}X)^{+}W^{\frac {1}{2}}\|_{2} where P \mathcal {P} is a set of diagonally dominant positive semidefinite matrices, by using a signature decomposition of weighting matrices W W and by applying the Binet-Cauchy formula and Cramer’s rule for determinants. The results are also extended to equality constrained linear least squares problems. In this paper we extend Forsgren’s results to a general complex matrix X X to establish several equivalent formulae for sup W ∈ P ‖ ( W 1 2 X ) + W 1 2 ‖ 2 \underset {W\in \mathcal {P} }{\text {sup}}\|(W^{\frac {1}{2}}X)^{+}W^{\frac {1}{2}}\|_{2} , where P \mathcal {P} is a set of diagonally dominant positive semidefinite matrices, or a set of weighting matrices arising from solving equality constrained least squares problems. We also discuss the stability property of these weighted pseudoinverses.