Upper perturbation bounds of weighted projections, weighted and constrained least squares problems

At each iteration step for solving mathematical programming and constrained optimization problems by using interior-point methods, one often needs to solve the weighted least squares (WLS) problem min(x is an element of Rn) parallel to W-1/2 (Ax + b)parallel to, or the weighted and constrained least squares (WLSE) problem min(x is an element of Rn) parallel to W-1/2 (Kx - g)parallel to subject to Lx = h, where W = diag(w(1),..., w(l)) >0 in which some w(i) --> + infinity and some w(i) --> 0. In this paper we will derive upper perturbation bounds of weighted projections associated with the WLS and WLSE problems when W ranges over the set D of positive diagonal matrices. We then apply these bounds to deduce upper perturbation bounds of solutions of WLS and WLSE problems when W ranges over D. We also extend the estimates to the cases when W ranges over a subset of real symmetric positive semidefinite matrices.21393195