EQUIVALENT FORMULAS FOR THE SUPREMUM AND STABILITY OF WEIGHTED PSEUDOINVERSES

During recent decades, there have been a great number of research arti cles studying interior-point methods for serving problems in mathemati cal programming and constrained optimization. Stewart and O'Leary obta ined an upper bound for scaled pseudoinverses sup(W epsilon P)parallel to((WX)-X-1/2)(+)W(1/2)parallel to(2) of a matrix X where P is a set of diagonal positive definite matrices. We improved their results to o btain 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 epsilo n P)parallel to((WX)-X-1/2)(+)W(1/2)parallel to(2) where P is a set of diagonally dominant positive semidefinite matrices; by using a signat ure decomposition of weighting matrices W and by applying the Binet-Ca uchy formula and Cramer's rule for determinants. The results are also extended to equality constrained linear least squares problems. In thi s paper we extend Forsgren's results to a general complex matrix X to establish several equivalent formulae for sup(W epsilon P)parallel to( (WX)-X-1/2)(+)W(1/2)parallel to(2) where P is a set of diagonally domi nant positive semidefinite matrices, or a set of weighting matrices ar ising from solving equality constrained least squares problems. We als o discuss the stability property of these weighted pseudoinverses.