Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods

Bocanegra, Silvana; Campos, Frederico Fonseca; Oliveira, Aurélio Ribeiro Leite de

doi:10.1007/s10589-006-9009-5

Cited by 52 publications

(93 citation statements)

References 29 publications

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“…Os experimentos foram realizados utilizando a versão modificada do PCx [3], nesta versão, o método direto usado para a solução dos sistemas lineares foi substituído por um método iterativo [1].…”

Section: Resultados Numéricosunclassified

Aprimoramento de um precondicionador híbrido aplicado ao método de pontos interiores

Oliveira¹,

Castro²,

Heredia³

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Neste trabalho propõe-se uma modificação nos parâmetros do precondicionador fatoração controlada de Cholesky e na escolha da base do precondicionador separador com o objetivo de aprimorar o cálculo da direção de busca do método de pontos interiores primaldual via o métodos iterativos precondicionados em duas fases. Nas iterações iniciais usa-se a fatoração controlada de Cholesky e em uma segunda fase o precondicionador separadoré acionado. Os parâmetros que controlam o preenchimento e a correção das falhas que ocorrem na diagonal são modificados para reduzir o número de reinícios da fatoração durante a construção da fatoração controlada de Cholesky. O cálculo dos novos parâmetrosé feito considerando a relação que existe entre as componentes da fatoração controlada de Cholesky obtida antes e depois da falha na diagonal. Este trabalho também considera o cálculo de uma base esparsa para o precondicionador separador com uma ordenação adequada das colunas da matriz de restrições do problema. Adicionalmente, apresenta-se um resultado teórico que mostra que, com o ordenamento proposto o número de condição da matriz precondicionada com o precondicionador separadoré limitado uniformemente por uma quantidade que depende apenas dos dados originais do problema. Experimentos numéricos com problemas de grande porte corroboram a robustez e eficiência computacional desta abordagem.Palavras-chave. Métodos de pontos interiores, Fatoração controlada de Cholesky, Número de condição, Precondicionador separador. IntroduçãoO maior esforço computacional do método de pontos interiores (MPI) consiste no cálculo da direção de busca queé obtida resolvendo sistemas lineares. A direção de busca pode ser calculada resolvendo opcionalmente dois sistemas lineares; o sistema aumentado e o sistema de equações normais (SEN). Neste trabalho o SENé resolvido usando uma abordagem de precondicionamento híbrido. As iterações iniciais usam o precondicionador fatoração 1

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Section: Resultados Numéricosunclassified

Aprimoramento de um precondicionador híbrido aplicado ao método de pontos interiores

Oliveira¹,

Castro²,

Heredia³

2018

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…A predictor direction x k , ỹ k , z k or the affine-scaling directions calculated from the system (4). This system of equations results from the Newton's method applied to the system (3).…”

Section: Primal-dual Interior Points Methodsmentioning

confidence: 99%

“…From tests carried out (Bocanegra et al, 2007), it has been proved that this factorization presents good results in the first iterations of the interior-point method, however it may deteriorates itself in the last ones, as the matrix Q gets very ill-conditioned.…”

Section: Controlled Cholesky Factorizationmentioning

confidence: 99%

“…Based on tests performed by Bocanegra et al (2007) it has been proved that these two preconditioners determine a different behavior for the conjugate gradients method in the solution of the linear systems which arise from interior-point methods. At the initial iterations of the interiorpoint method, the method of the conjugate gradients solves the linear systems involved more efficiently using the preconditioner obtained by the controlled Cholesky factorization (Campos & Birkett, 1998).…”

Section: Introductionmentioning

confidence: 99%

“…Moreover, improvements to the efficiency of the splitting preconditioner are presented, consequently decreasing the number of iterations of the conjugate gradients method. The results are presented using large-scale problems and comparing the results with the ones obtained by Bocanegra et al (2007).…”

Section: Introductionmentioning

confidence: 99%

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Heuristics for implementation of a hybrid preconditioner for interior-point methods

2011

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ABSTRACT. This article presents improvements to the hybrid preconditioner previously developed for the solution through the conjugate gradient method of the linear systems which arise from interior-point methods. The hybrid preconditioner consists of combining two preconditioners: controlled Cholesky factorization and the splitting preconditioner used in different phases of the optimization process. The first, with controlled fill-in, is more efficient at the initial iterations of the interior-point methods and it may be inefficient near a solution of the linear problem when the system is highly ill-conditioned; the second is specialized for such situation and has the opposite behavior. This approach works better than direct methods for some classes of large-scale problems. This work has proposed new heuristics for the integration of both preconditioners, identifying a new change of phases with computational results superior to the ones previously published. Moreover, the performance of the splitting preconditioner has been improved through new orderings of the constraint matrix columns allowing savings in the preconditioned conjugate gradient method iterations number. Experiments are performed with a set of large-scale problems and both approaches are compared with respect to the number of iterations and running time.

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A new preconditioning approach for an interior point‐proximal method of multipliers for linear and convex quadratic programming

Bergamaschi

Gondzio

Martínez

et al. 2021

Numerical Linear Algebra App

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In this article, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers, which in turn results in a primal-dual regularized interior point method. Application of this method gives rise to a sequence of increasingly ill-conditioned linear systems which cannot always be solved by factorization methods, due to memory and CPU time restrictions. We propose a novel preconditioning strategy which is based on a suitable sparsification of the normal equations matrix in the linear case, and also constitutes the foundation of a block-diagonal preconditioner to accelerate MINRES for linear systems arising from the solution of general quadratic programming problems. Numerical results for a range of test problems demonstrate the robustness of the proposed preconditioning strategy, together with its ability to solve linear systems of very large dimension.

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Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods

Cited by 52 publications

References 29 publications

Aprimoramento de um precondicionador híbrido aplicado ao método de pontos interiores

Aprimoramento de um precondicionador híbrido aplicado ao método de pontos interiores

Heuristics for implementation of a hybrid preconditioner for interior-point methods

A new preconditioning approach for an interior point‐proximal method of multipliers for linear and convex quadratic programming

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