Preprint 1-2006

On the Efficient Update of Rectangular LU Factorizations subject to Low Rank Modifications

Author(s) : Peter Stange , Andreas Griewank , Matthias Bollhöfer

Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 1-2006

MSC 2000

65F05 Direct methods for linear systems and matrix inversion

65K05 Mathematical programming

Abstract :
In this paper we introduce a new method for the computation of KKT matrices that arise from solving constrained, nonlinear optimization problems. This method requires updating of null-space factorizations after a low rank modification. The update procedure has the advantage that it is significantly cheaper than a re-factorization of the system at each new iterate. This paper focuses on the cheap update of a rectangular LU decomposition after a rank-1 modification. Two different procedures for updating the LU factorization are presented in detail and compared regarding their costs of computation and their stability. Moreover we will introduce an extension of these algorithms which further improves the computation time. This turns out to be an excellent alternative to algorithms based on orthogonal transformations.

Keywords : KKT-System, quasi-Newton, updating factorization, LU decomposition