Sistemas Lineares Aproximados Derivados de Problemas de Fluxo Multiproduto em Métodos de Pontos Interiores

Authors

  • Luciana Yoshie Tsuchiya Instituto Federal do Paraná
  • Aurelio Ribeiro Leite de Oliveira Universidade Estadual de Campinas

DOI:

https://doi.org/10.5540/tema.2017.018.01.0139

Keywords:

Método de pontos interiores primal-dual, Fatoração de Cholesky, Sistema de Equações Normais.

Abstract

Uma das abordagens utilizadas para resolver o sistema linear que surge a cada iteração nos métodos de pontos interiores primal-dual é reduzi-lo a um sistema linear equivalente simétrico definido positivo, conhecido como sistema de equações normais, e aplicar a fatoração de Cholesky na matriz do sistema.

A grande  desvantangem desta abordagem é o preenchimento gerado durante a fatoração, o que pode tornar seu uso inviável, por limitação de tempo e memória. Com o intuito de contornar o problema de preenchimento gerado na fatoração de Cholesky, neste trabalho, estamos propondo uma abordagem que resolve de forma direta sistemas lineares aproximados do sistema de equações normais derivados de problemas de fluxo multiproduto e que exerce um certo controle sobre o preenchimento.

References

G. Al-Jeiroudi and J. Gondzio, About the “Convergence Analysis of the Inexact Infeasible Interior-Point Method for Linear Optimization”, Journal of Optimization Theory and Applications, 141 (2009), pp. 231–247.

V. Baryamureeba and T. Steihaug, “On the convergence of an inexact primaldual interior point method for linear programming”, in Lecture Notes in Computer Science, Springer, Berlin/Heidelberg, 2006.

S. Bellavia, “An inexact interior point method”, Journal of Optimization Theory and Applications, 96 (1988), pp. 199–121.

M. Benzi, “Preconditioning techniques for large linear systems: A survey, Journal of Computational Physics”, 182, No 2 (2002), pp. 418–477.

S. Bocanegra, F. F. Campos, and A. R. L. Oliveira, “Using a hybrid preconditioner for solving large-scale linear systems arising from interior point methods”, Computational Optimization and Applications, 36 (2007), pp. 149–164. Special issue on “Linear Algebra Issues arising in Interior Point Methods”.

F. F. Campos, “Analysis of Conjugate Gradients - type methods for solving linear equations”, PhD thesis, Oxford University Computing Laboratory, Oxford, 1995.

J. Czyzyk, S. Mehrotra, S. J. Wright, and M. Wagner, “PCx: An interior-point code for linear programming”, Optimization Methods and Software, 11, No 1 (1999), pp. 397–430.

I. S. Duff and G. A. Meurant, “The effect of ordering on preconditioned conjugate gradients”, BIT, 29 (1989), pp. 635–657.

C. T. L. S. Ghidini, A. R. L. Oliveira, and D. C. Sorensen, “Computing a hybrid preconditioner approach to solve the linear systems arising from interior point methods for linear programming using the gradient conjugate method”, Annals of Management Science, (2013).

N. Highan, “Acuraccy and Stability of Numerical Algorithms”, SIAM Publications, Philadelphia, 1996.

J. L. Kennington, S. Niemi, and S. J. Wichmann, “An empirical evaluation of the korbx algorithms for military airlift applications”, Operations Research, 28, pp. 240–248.

M. Kojima, S. Mizuno, and A. Yoshise, “A primal-dual interiorpoint method

for linear programming. progress in mathematical programming”, interior-point

and related methods, scSpring-Verlag, New York, (1989), pp. 29– 47.

J. Korzak, Convergence analysis of inexact infeasible-interior-point algorithms for solving linear programming problems”, Siam J. Optim, 11 (2000), pp. 133– 148.

T. A. Manteuffel, “An incomplete factorization technique for positive definite linear systems”, Math. Comp., 34 (1980), pp. 473–497.

R. D. S. Monteiro and J. W. O’Neal, “Convergence analysis of long-step primaldual infeasible interior point LP algorithm based on iterative linear solvers”, Georgia Institute of Technology, 2003.

E. Ng and B. P. Peyton, “Block sparse Cholesky algorithms on advanced uniprocessors computers”, SIAM Journal on Scientific Stat. Computing, 14 (1993), pp. 1034–1056.

R. J. Vanderbei, “Linear Programming - Foundations and Extensions”, Kluwer Academics Publishers, Boston, USA, 1996.

M. I. Velazco, A. R. L. Oliveira, and F. F. Campos, “Heuristics for implementation of a hybrid preconditioner for interior-point methods”, Pesquisa Operacional, 34 (2011), pp. 2553–2561.

S. J.Wright, “Primal-Dual Interior-Point Methods”, SIAM Publications, SIAM, Philadelphia, PA, USA, 1996.

Downloads

Additional Files

Published

2017-05-22

How to Cite

Tsuchiya, L. Y., & Oliveira, A. R. L. de. (2017). Sistemas Lineares Aproximados Derivados de Problemas de Fluxo Multiproduto em Métodos de Pontos Interiores. Trends in Computational and Applied Mathematics, 18(1), 139. https://doi.org/10.5540/tema.2017.018.01.0139

Issue

Vol. 18 No. 1 (2017)

Section

Original Article

License

Copyright

Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.

Intellectual Property and Terms of Use

The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.

The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.