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Using constraint preconditioners with regularized saddle-point problems

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Abstract

The problem of finding good preconditioners for the numerical solution of a certain important class of indefinite linear systems is considered. These systems are of a 2 by 2 block (KKT) structure in which the (2,2) block (denoted by -C) is assumed to be nonzero.

In Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl. 21 (2000), Keller, Gould and Wathen introduced the idea of using constraint preconditioners that have a specific 2 by 2 block structure for the case of C being zero. We shall give results concerning the spectrum and form of the eigenvectors when a preconditioner of the form considered by Keller, Gould and Wathen is used but the system we wish to solve may have C ≠0. In particular, the results presented here indicate clustering of eigenvalues and, hence, faster convergence of Krylov subspace iterative methods when the entries of C are small; such a situations arise naturally in interior point methods for optimization and we present results for such problems which validate our conclusions.

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Authors and Affiliations

  1. Computational Science and Engineering Department, Rutherford Appleton Laboratory, Chilton, Oxfordshire, OX11 0QX, England, UK

    H. S. Dollar & N. I. M. Gould

  2. Numerical Analysis Group, Oxford University Computing Laboratory, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK

    N. I. M. Gould & A. J. Wathen

  3. Design Methods and Solutions, NXP Semiconductors, High Tech Campus–48, 5656 AE, Eindhoven, The Netherlands

    W. H. A. Schilders

  4. Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, PO Box 513, 5600 MB, Eindhoven, The Netherlands

    W. H. A. Schilders

Authors
  1. H. S. Dollar
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  2. N. I. M. Gould
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  3. W. H. A. Schilders
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  4. A. J. Wathen
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Correspondence to H. S. Dollar.

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Dollar, H.S., Gould, N.I.M., Schilders, W.H.A. et al. Using constraint preconditioners with regularized saddle-point problems. Comput Optim Appl 36, 249–270 (2007). https://doi.org/10.1007/s10589-006-9004-x

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