An iterative criterion for H-Matrices

Li, Bishan; Li, Lei; Harada, Masunori; Niki, Hiroshi; Tsatsomeros, Michael J.

doi:10.1016/s0024-3795(97)00267-x

Cited by 48 publications

(7 citation statements)

References 4 publications

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“…The sets of diagonally dominant matrices and scaled diagonally dominant matrices are known to be cones contained in S n + (see e.g., [2,22] for details). The inclusive relation among them has been studied in, e.g., [7,8]. Ahmadi et al [1,2] used these sets as initial approximations of their cutting plane method.…”

Section: Introductionmentioning

confidence: 99%

Polyhedral approximations of the semidefinite cone and their application

2021

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We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (Ann Oper Res 265:155–182, 2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

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Section: Introductionmentioning

confidence: 99%

Polyhedral approximations of the semidefinite cone and their application

2021

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“…The sets of diagonally dominant matrices and scaled diagonally dominant matrices are known to be cones contained in S n + , (see, e.g., [22] and [1] for details). The inclusive relation among them has been studied in, e.g., [7] and [8]. Ahmadi et al [1] and [2] used these sets as initial approximations of their cutting plane method.…”

Section: Introductionmentioning

confidence: 99%

Polyhedral approximations of the semidefinite cone and their application

Wang,

Tanaka,

Yoshise

2019

Preprint

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We develop techniques to construct a series of sparse polyhedral approximations of the semidefinite cone. Motivated by the semidefinite (SD) bases proposed by Tanaka and Yoshise (2018), we propose a simple expansion of SD bases so as to keep the sparsity of the matrices composing it. We prove that the polyhedral approximation using our expanded SD bases contains the set of all diagonally dominant matrices and is contained in the set of all scaled diagonally dominant matrices. We also prove that the set of all scaled diagonally dominant matrices can be expressed using an infinite number of expanded SD bases. We use our approximations as the initial approximation in cutting plane methods for solving a semidefinite relaxation of the maximum stable set problem. It is found that the proposed methods with expanded SD bases are significantly more efficient than methods using other existing approximations or solving semidefinite relaxation problems directly.

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“…Algoritmo de Li, Li, Harada, Niki y Tsatsomeros. El algoritmo presentado en [30] por Li, Li, Harada, Niki y Tsatsomeros es el primero que incluye un algoritmo detallado (Algoritmo H) para la determinación de Hmatrices. Su desarrollo se dice que es un intento de mejora del algoritmo de [24], pero el resultado está en la línea de lo que hemos indicado anteriormente como común objetivo de este tipo de algoritmos: encontrar una matriz diagonal D tal que, o AD es SDD, o AD no cumple la dominancia diagonal en ninguna de sus filas.…”

Section: Algunos Algoritmos Previosunclassified

“…El Algoritmo H hace uso de un parámetro > 0 que se ha de introducir, así como la matriz A ∈ C n×n a estudiar. Indicamos a continuación el algoritmo tal y como viene en [30].…”

Section: Algunos Algoritmos Previosunclassified

“…Desde hace tiempo se han realizado diversos intentos para conseguir la determinación automática de las H-matrices. En ese sentido, en diversos trabajos (ver por ejemplo [22], [24], [30], [29], [31] y [35]) se recogen algunos intentos para establecer si una matriz dada A es, o no es, una H-matriz a partir de la condición de ser GSDD, esto es, el proceso se basa en encontrar iterativamente una matriz diagonal D de forma que, si M representa la matriz de comparación de la matriz original A, entonces se obtenga que M D es SDD o, recíprocamente, M D no consigue la dominancia diagonal en ninguna de sus filas. Así pues, este tipo de algoritmo no es aplicable a El análisis del carácter SDD de una matriz tampoco es consistente con matrices reducibles.…”

Section: Introductionunclassified

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Determinación y propiedades de H-matrices

Guilleard¹,

Antonio²

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The essential topic of this memory is the study of H-matrices as they were introduced by Ostrowski and hereinafter extended and developed by different authors.In this study three slopes are outlined: 1) the iterative or automatic determination of H-matrices, 2) the properties inherent in the H-matrices and 3) the matrices related to H-matrices. H-matrices acquire every time major relevancy due to the fact that they arise in numerous applications so much in Mathematics, since in the Industry between. Between these applications we can mention the following ones: 1) in the discretization of certain parabolic non-linear equations, 2) in the system resolution of linear equations, assuring his presence the convergence of iterative classic methods and 3) in the resolution of problems of free contour in Analysis of Fluids.

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An iterative criterion for H-Matrices

Cited by 48 publications

References 4 publications

Polyhedral approximations of the semidefinite cone and their application

Polyhedral approximations of the semidefinite cone and their application

Polyhedral approximations of the semidefinite cone and their application

Determinación y propiedades de H-matrices

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