A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories

We present a class of incomplete orthogonal factorization methods based on Givens rotations for large sparse unsymmetric matrices. These methods include: Incomplete Givens Orthogonalization (IGO-method) and its generalisation (GIGO-method), which drop entries from the incomplete orthogonal and upper triangular factors by position; Threshold Incomplete Givens Orthogonalization (TIGO()-method), which drops entries dynamically by their magnitudes; and its generalisation (GTIGO(,p)-method), which drops entries dynamically by both their magnitudes and positions. Theoretical analyses show that these methods can produce a nonsingular sparse incomplete upper triangular factor and either a complete orthogonal factor or a sparse nonsingular incomplete orthogonal factor for a general nonsingular matrix. Therefore, these methods can potentially generate efficient preconditioners for Krylov subspace methods for solving large sparse systems of linear equations. Moreover, the upper triangular factor is an incomplete Cholesky factorization preconditioner for the normal equations matrix from least-squares problems.     

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.     

一类全局收敛的记忆梯度法及其线性收敛性

本文研究一类新的解无约束最优化问题的记忆梯度法,在强Wolfe线性搜索下证明了其全局收敛性．当目标函数为一致凸函数时,对其线性收敛速率进行了分析．数值试验表明算法是很有效的．