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

Abstract

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.

Theoretical and computational issues for improving the performance of linear optimization methods

“…This study extends the earlier investigations regarding this method, which were presented by Arenales (1984), Sousa et al (2005) and Silva et al (2007). In particular, the main contributions are:…”

“…First, we presented a novel theoretical discussion of the variant, which extends the preliminary studies of Arenales (1984), Sousa et al (2005) and Silva et al (2007). Then, we discussed the main techniques that are useful to obtain an ecient and stable implementation of the method.…”

“…Hence, we describe how to extend the LU factorization and LU update schemes proposed by Suhl (1990, 1993) to the particular case we are dealing with (Section 3.6.2). These techniques are known to be more ecient than those used in the computational implementations of Sousa et al (2005) and Silva et al (2007). They typically reduce the ll-in of the sparse representation of the basic matrix, and improve the numerical stability of the method.…”