Constraint Preconditioning for Indefinite Linear Systems

Keller, Carsten; Gould, Nicholas I. M.; Wathen, Andrew J.

doi:10.1137/s0895479899351805

Cited by 298 publications

(260 citation statements)

References 16 publications

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“…Matrix P of (3) belongs to a wide class of block-preconditioners that have been studied in numerous applications in the context of partial differential equations, see for example [15,24] and the references therein, and in the context of optimization [11,14,19,21,22]. We perform the spectral analysis of P −1 H and conclude, as in [19], that for convex optimization problems all the eigenvalues of this matrix are strictly positive.…”

Section: Introductionmentioning

confidence: 96%

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Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

120

128

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Abstract. Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.

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Section: Introductionmentioning

confidence: 96%

“…We perform the spectral analysis of P −1 H and conclude, as in [19], that for convex optimization problems all the eigenvalues of this matrix are strictly positive. This feature offers much promise for computations.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

120

128

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“…In this section we consider a constraint preconditioning approach, see [10]. Although this approach has been used previously in the context of mixed approximations, our strategy is actually quite different -it is tailored to the special structure of the coefficient matrix in (1.8).…”

Section: A Constraint Preconditionermentioning

confidence: 99%

“…The preconditioned matrix is not diagonalisable (it only has n I + 2n B − nr linearly independent eigenvectors, see [10,Thm. 2.3]).…”

Section: Proposition 33 the Matrix P In (31) Is Non-singularmentioning

confidence: 99%

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Silvester

Mihajlović

2004

BIT Numerical Mathematics

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Abstract.We examine the convergence characteristics of a preconditioned Krylov subspace solver applied to the linear systems arising from low-order mixed finite element approximation of the biharmonic problem. The key feature of our approach is that the preconditioning can be realized using any "black-box" multigrid solver designed for the discrete Dirichlet Laplacian operator. This leads to preconditioned systems having an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. Numerical results show that the performance of the methodology is competitive with that of specialized fast iteration methods that have been developed in the context of biharmonic problems. (2000): 65F10, 65N12, 65N22, 65N55. AMS subject classification

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“…Numerous solution methods for the saddle point systems of the form (1) can be found in the literature and many of them have focused on preconditioning techniques for Krylov subspace iterative solvers [1,2,3,7,12,18,23,24,26,28,31]. As a direct method against iterative solvers, various techniques on symmetric indefinite factorization P TÅ P = LDL T can be found in [9,14,21,34,35,38], where P is a permutation matrix, L is unit lower triangular matrix, D is block-diagonal matrix with blocks of order 1 or 2.…”

Section: Introductionmentioning

confidence: 99%

Sparse block factorization of saddle point matrices

Lungten

Schilders

Maubach

2016

Linear Algebra and its Applications

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The factorization method presented in this paper takes advantage of the special structures and properties of saddle point matrices. A variant of Gaussian elimination equivalent to the Cholesky's factorization is suggested and implemented for factorizing the saddle point matrices block-wise with small blocks of order 1 and 2. The Gaussian elimination applied to these small blocks on block level also induces a block 3 × 3 structured factorization of which the blocks have special properties. We compare the new block factorization with the Schilders' factorization in terms of the sparsity of their factors and computational efficiency. The factorization can be used as a direct method, and also anticipate for preconditioning techniques.

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Constraint Preconditioning for Indefinite Linear Systems

Cited by 298 publications

References 16 publications

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

A Black-Box Multigrid Preconditioner for the Biharmonic Equation

Sparse block factorization of saddle point matrices

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