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IMA Journal of Numerical Analysis 2008 28(4):806-826; doi:10.1093/imanum/drn058
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Numerically stable LDLT factorizations in interior point methods for convex quadratic programming

D. Goldfarb{dagger}

Industrial Engineering and Operations Research, Department, Columbia University, New York, NY 10027, USA

K. Scheinberg{ddagger}

IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA

{dagger} Corresponding author. Email: goldfarb{at}columbia.edu

{ddagger} Email: katyas{at}us.ibm.com

Received on 3 April 2007. Accepted for publication 6 September 2008.


  Abstract

Fletcher & Powell (1974, Math. Comput., 28, 1067–1087) proposed a numerically stable method for updating the LDLT factorization of a symmetric positive-definite matrix when a symmetric low-rank term is added to it. In Goldfarb & Scheinberg (2004, Math. Program., 99, 1–34) we proposed a product-form version of the method of Fletcher and Powell for use in interior point methods for linear programming and studied its numerical stability. In this paper we extend these results to convex quadratic programming where the Hessian matrix of the objective function is, or can be approximated by, a diagonal matrix plus a matrix of low rank. Our new results are based on showing that the elements of the unit lower triangular matrix in the LDLT factorizations that arise in this context are uniformly bounded as the duality gap is driven to zero. Practicable versions of our approach are described for structured quadratic programs that arise in support vector machines and portfolio optimization.

Key Words: product-form LDLT factorization; Cholesky factorization; convex quadratic programming; interior point methods; normal equations; support vector machines; portfolio optimization

Dedicated to Prof. M. J. D. Powell on the occasion of his 70th birthday.


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