Solution of large‐scale weighted least‐squares problems

A sequence of least‐squares problems of the form min_y∥G^1/2(A^T y?h)∥₂, where G is an n×n positive‐definite diagonal weight matrix, and A an m×n (m?n) sparse matrix with some dense columns; has many applications in linear programming, electrical networks, elliptic boundary value problems, and structural analysis. We suggest low‐rank correction preconditioners for such problems, and a mixed solver (a combination of a direct solver and an iterative solver). The numerical results show that our technique for selecting the low‐rank correction matrix is very effective. Copyright © 2002 John Wiley & Sons, Ltd.     

The Impact of Equal Weighting of Low- and High-Confidence Observations on Robust Linear Regression Computations

Equal weighting of low- and high-confidence observations occurs for Huber, Talwar, and Barya weighting functions when Newton's method is used to solve robust linear regression problems. This leads to easy updates and/or downdates of existing matrix factorizations or easy computation of coefficient matrices in linear systems from previous ones. Thus Newton's method based on these functions has been shown to be computationally cheap. In this paper we show that a combination of Newton's method and an iterative method is a promising approach for solving robust linear regression problems. We show that Newton's method based on the Talwar function is an active set method. Further we show that it is possible to obtain improved estimates of the solution vector by combining a line search method like Newton's method with an active set method.This revised version was published online in October 2005 with corrections to the Cover Date.