Numerical Algorithms

, Volume 70, Issue 1, pp 9–41 | Cite as

Limited-memory LDL factorization of symmetric quasi-definite matrices with application to constrained optimization

Original Paper


We propose a generalization of the limited-memory Cholesky factorization of Lin and Moré (SIAM J. Sci. Comput. 21(1), 24–45, 1999) to the symmetric indefinite case with special interest in symmetric quasi-definite matrices. We use this incomplete factorization to precondition two formulations of linear systems arising from regularized interior-point methods for quadratic optimization. An advantage of the limited-memory approach is predictable memory requirements. We establish existence of incomplete factors when the input matrix is an H-matrix but our numerical results illustrate that the factorization succeeds more generally. An appropriate diagonal shift is applied whenever the input matrix is not quasi definite. As the memory parameter increases an efficiency measure of the preconditioner suggested by Scott and Tůma (2013) improves. The combination of the 3×3 block formulation analyzed by Greif, Moulding, and Orban (SIAM J. Optim. 24(1), 49–83, 2014), the SYMAMD ordering, and a moderate memory parameter results in encouraging performance.


Preconditioning Symmetric quasi definite Incomplete factorization Limited-memory factorization Interior-point methods 

Mathematics Subject Classifications (2000)

15A06 15A23 15B57 90C06 90C20 65F08 65F10 65F22 65F50 


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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.GERAD and Department of Mathematics and Industrial EngineeringÉcole PolytechniqueMontréalCanada

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