A new class of preconditioners for large-scale linear systems from interior point methods for linear programming

Oliveira, Aurélio Ribeiro Leite de; Sorensen, Danny C.

doi:10.1016/j.laa.2004.08.019

Cited by 90 publications

(145 citation statements)

References 24 publications

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“…Such preconditioners are widely used in linear systems obtained from a discretization of partial differential equations [15,24]. The preconditioners for the augmented system have also been used in the context of linear programming [14,23] and in the context of nonlinear programming [11,19,21,22,25]. As was shown in [23], the preconditioners for indefinite augmented system offer more freedom than those for the normal equations.…”

Section: Preconditionersmentioning

confidence: 99%

“…The preconditioners for the augmented system have also been used in the context of linear programming [14,23] and in the context of nonlinear programming [11,19,21,22,25]. As was shown in [23], the preconditioners for indefinite augmented system offer more freedom than those for the normal equations. Moreover, the factorization of the augmented system is sometimes much easier than that of the normal equations [2] (this is the case, for example, when A contains dense columns).…”

Section: Preconditionersmentioning

confidence: 99%

See 1 more Smart Citation

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

119

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: Preconditionersmentioning

confidence: 99%

Section: Preconditionersmentioning

confidence: 99%

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Bergamaschi

Gondzio

Zilli

2004

Computational Optimization and Applications

119

128

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“…Unfortunately, for iterative (Krylov-subspace) methods the ill-conditioning of Θ and the resulting ill-conditioning of (2) makes the system intractable unless appropriately preconditioned [7,21,29,30]. We formally state this observation below.…”

Section: Kkt System In Interior Point Methodsmentioning

confidence: 99%

“…Finally, to get the last remaining bound, we consider again (30) and forλ > 0 take the largest possible terms on its left-hand-side to get…”

Section: Regularized Kkt Systemmentioning

confidence: 99%

Matrix-free interior point method

Gondzio

2010

Comput Optim Appl

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In this paper we present a redesign of a linear algebra kernel of an interior point method to avoid the explicit use of problem matrices. The only access to the original problem data needed are the matrix-vector multiplications with the Hessian and Jacobian matrices. Such a redesign requires the use of suitably preconditioned iterative methods and imposes restrictions on the way the preconditioner is computed. A two-step approach is used to design a preconditioner. First, the Newton equation system is regularized to guarantee better numerical properties and then it is preconditioned. The preconditioner is implicit, that is, its computation requires only matrix-vector multiplications with the original problem data. The method is therefore well-suited to problems in which matrices are not explicitly available and/or are too large to be stored in computer memory. Numerical properties of the approach are studied including the analysis of the conditioning of the regularized system and that of the preconditioned regularized system. The method has been implemented and preliminary computational results for small problems limited to 1 million of variables and 10 million of nonzero elements demonstrate the feasibility of the approach.

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“…Furthermore, it has been noted in [66] that every preconditioner for the condensed system induces a preconditioner for the KKT system, while the converse is not true. In this context, first esperiments have been performed using symmetric Krylov subspace solvers, such as SYMMLQ coupled with suitable symmetric preconditioners [38] and SQMR, with a modified SSOR preconditioner [34].…”

Section: Preconditioning the Kkt Systemmentioning

confidence: 99%

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

2008

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The solution of KKT systems is ubiquitous in optimization methods and often dominates the computation time, especially when large-scale problems are considered. Thus, the effective implementation of such methods is highly dependent on the availability of effective linear algebra algorithms and software, that are able, in turn, to take into account specific needs of optimization. In this paper we discuss the mutual impact of linear algebra and optimization, focusing on interior point methods and on the iterative solution of the KKT system. Three critical issues are addressed: preconditioning, termination control for the inner iterations, and inertia control.

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A new class of preconditioners for large-scale linear systems from interior point methods for linear programming

Cited by 90 publications

References 24 publications

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Matrix-free interior point method

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

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