Computing Sparse LU Factorizations for Large-Scale Linear Programming Bases

Suhl, Uwe H.; Suhl, Leena

doi:10.1287/ijoc.2.4.325

Cited by 87 publications

(37 citation statements)

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“…Este procedimento para obter estabilidade pode não ser muito útil se os elementos de B são de magnitudes extremamente diferentes (Suhl & Suhl, 1990). Quando isso ocorre, faz-se uma mudança de escala nos elementos da matriz.…”

Section: Exemplounclassified

Métodos tipo dual simplex para problemas de otimização linear canalizados e esparsos

2007

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ResumoOs problemas de otimização linear canalizados e esparsos, objeto principal deste trabalho, surgem em várias aplicações, como por exemplo, problemas de planejamento da produção, problemas de mistura, entre outras. Métodos tipo dual simplex com busca linear por partes foram propostos e analisados em Sousa et al. (2005), com resultados efetivos para problemas densos pequenos e agora são analisados para problemas esparsos maiores. Algumas heurísticas de pivotamento foram implementadas para tentar manter a esparsidade e reduzir o tempo total de resolução dos problemas. Um conjunto de exemplos com estruturas esparsas que tipicamente ocorrem na prática foram gerados aleatoriamente para analisar o desempenho dos métodos. Os resultados computacionais demonstram a eficiência da abordagem.Palavras-chave: otimização linear; otimização linear por partes; dualidade; esparsidade. AbstractTwo-side constraint and sparse linear optimization problems, the main object of this work, appear in several applications, such as, production planning problems, mix problems among others. Dual Simplex-typed methods, called two-side constraint dual simplex methods with piecewise linear search were proposed and analyzed in Sousa et al. (2005), which showed effective results for dense and small problems and now they are analyzed for larger and sparse problems. These methods were implemented together with some pivoting heuristics to maintain sparsity and to reduce the running time. Sets of linear optimization randomly generated problems with sparse structures that occur in real world were used to analyze the performance of the methods. The computational results show the efficiency of the approaches.

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Section: Exemplounclassified

Métodos tipo dual simplex para problemas de otimização linear canalizados e esparsos

2007

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“…The first of these is dynamic LUfactorization using Markowitz threshold pivoting. This approach was perfected by Suhl and Suhl (1990), and has become a standard part of modern codes. In previousgeneration codes, "preassigned pivot" sequences were used in the numerical factorization (see Hellerman and Rarick 1971).…”

Section: Algorithmic Improvementsmentioning

confidence: 99%

Solving Real-World Linear Programs: A Decade and More of Progress

Bixby

2002

Operations Research

304

130

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This paper is an invited contribution to the 50th anniversary issue of the journal Operations Research, published by the Institute of Operations Research and Management Science (INFORMS). It describes one person's perspective on the development of computational tools for linear programming. The paper begins with a short personal history, followed by historical remarks covering the some 40 years of linear-programming developments that predate my own involvement in this subject. It concludes with a more detailed look at the evolution of computational linear programming since 1987.

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“…Klee and Minty [67] constructed a problem of dimension n, the solution of which requires 2 n iterations of the simplex method, which demonstrates that the simplex method is not a polynomial algorithm. It has to be said, however, that it is only a theoretical drawback and in practice it is rather exceptional for the simplex method to perform more than m + n iterations on its way to an optimal solution [13,33,54,75,98].…”

Section: Interior Point Methods: Backgroundmentioning

confidence: 99%

“…It is worth mentioning at this point that the presence of interior point methods have put considerable pressure on developers of commercial simplex implementations and have led to impressive developments of the simplex method over the last 25 years [13,33,54,75,98]. Both methods are widely used nowadays and continue to compete with each other.…”

Section: Introductionmentioning

confidence: 99%

Interior point methods 25 years later

Gondzio

2012

European Journal of Operational Research

307

259

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Interior point methods for optimization have been around for more than 25 years now. Their presence has shaken up the field of optimization. Interior point methods for linear and (convex) quadratic programming display several features which make them particularly attractive for very large scale optimization. Among the most impressive of them are their lowdegree polynomial worst-case complexity and an unrivalled ability to deliver optimal solutions in an almost constant number of iterations which depends very little, if at all, on the problem dimension. Interior point methods are competitive when dealing with small problems of dimensions below one million constraints and variables and are beyond competition when applied to large problems of dimensions going into millions of constraints and variables.In this survey we will discuss several issues related to interior point methods including the proof of the worst-case complexity result, the reasons for their amazingly fast practical convergence and the features responsible for their ability to solve very large problems. The ever-growing sizes of optimization problems impose new requirements on optimization methods and software. In the final part of this paper we will therefore address a redesign of interior point methods to allow them to work in a matrix-free regime and to make them well-suited to solving even larger problems.

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Computing Sparse LU Factorizations for Large-Scale Linear Programming Bases

Cited by 87 publications

References 0 publications

Métodos tipo dual simplex para problemas de otimização linear canalizados e esparsos

Métodos tipo dual simplex para problemas de otimização linear canalizados e esparsos

Solving Real-World Linear Programs: A Decade and More of Progress

Interior point methods 25 years later

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