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A Direct Orthogonal Sparse Static Methodology for a Finite Continuation Hybrid LP Solver

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Applied Parallel Computing. State of the Art in Scientific Computing (PARA 2004)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3732))

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Abstract

A finite continuation method for solving linear programs (LPs) has been recently put forward by K. Madsen and H. B. Nielsen which, to improve its performance, can be thought of as a Phase-I for a non-simplex active-set method (also known as basis-deficiency-allowing simplex variation); this avoids having to start the simplex method from a highly degenerate square basis. An efficient sparse implementation of this combined hybrid approach to solve LPs requires the use of the same sparse data structure in both phases, and a way to proceed in Phase-II when a non-square working matrix is obtained after Phase-I.

In this paper a direct sparse orthogonalization methodology based on Givens rotations and a static sparsity data structure is proposed for both phases, with a Linpack-like downdating without resorting to hyperbolic rotations. Its sparse implementation (recently put forward by us) is of reduced-gradient type, regularization is not used in Phase-II, and occasional refactorizations can take advantage of row orderings and parallelizability issues to decrease the computational effort.

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References

  1. Björck, Å., Duff, I.S.: A direct method for the solution of sparse linear least squares problems. Linear Algebra Appl. 34, 43–67 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  2. Edlund, O.: A software package for sparse orthogonal factorization and updating. ACM Trans. Math. Software 28(4), 448–482 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  3. Fletcher, R., Powell, M.J.D.: On themodification of LDLT factorizations. Math. Comp. 29, 1067–1087 (1974)

    MathSciNet  Google Scholar 

  4. Alan George, J., Heath, M.T.: Solution of sparse linear least squares problems using Givens rotations. Linear Algebra Appl. 34, 69–83 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alan George, J., Liu, J.W.-H.: An object-oriented approach to the design of a user interface for a sparse matrix package. SIAM J. Matrix Anal. Applics. 20(4), 953–969 (1999)

    Article  MATH  Google Scholar 

  6. Guerrero-García, P., Santos-Palomo, Á.: Solving a sequence of sparse compatible systems. Technical report MA-02-03, Department of Applied Mathematics, University of Málaga (November 2002) (Submitted for publication)

    Google Scholar 

  7. Guerrero-García, P., Santos-Palomo, Á.: A sparse orthogonal factorization technique for certain stiff systems of linear ODEs. In: Marín, J., Koncar, V. (eds.) Industrial Simulation Conference 2004 Proceedings, June 2004, pp. 17–19 (2004)

    Google Scholar 

  8. Madsen, K., Nielsen, H.B., Pinar, M.Ç.: A new finite continuation algorithm for linear programming. SIAM J. Optim. 6(3), 600–616 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  9. Maros, I.: Computational Techniques of the Simplex Method. Kluwer Academic Publishers, Dordrecht (2003)

    MATH  Google Scholar 

  10. Matstoms, P.: Sparse linear least squares problems in optimization. Comput. Optim. Applics. 7(1), 89–110 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Megiddo, N.: On finding primal- and dual-optimal bases. ORSA J. Computing 3(1), 63–65 (1991)

    MATH  MathSciNet  Google Scholar 

  12. Murty, K.G.: Linear Programming. John Wiley and Sons, Chichester (1983)

    MATH  Google Scholar 

  13. Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann (1988)

    Google Scholar 

  14. Nielsen, H.B.: AAFAC: A package of FORTRAN 77 subprograms for solving A T Ax = c. Tech. report, Institute for Numerical Analysis, Technical University of Denmark, Lyngby, Denmark (March 1990)

    Google Scholar 

  15. Nielsen, H.B.: Computing a minimizer of a piecewise quadratic. Tech. report, Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark (September 1998)

    Google Scholar 

  16. Nielsen, H.B.: Algorithms for linear optimization, 2nd edition. Tech. report, Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark (February 1999)

    Google Scholar 

  17. Pan, P.-Q.: A basis-deficiency-allowing variation of the simplexmethod for linear programming. Computers Math. Applic. 36(3), 33–53 (1998)

    Article  MATH  Google Scholar 

  18. Santos-Palomo, Á., Guerrero-García, P.: A non-simplex active-setmethod for linear programs in standard form. Stud. Inform. Control 14(2) (June 2005)

    Google Scholar 

  19. Santos-Palomo, Á., Guerrero-García, P.: Updating and downdating an upper trapezoidal sparse orthogonal factorization. Accepted for publication in IMA J. Numer. Anal.

    Google Scholar 

  20. Saunders, M.A.: Large-scale linear programming using theCholesky factorization. Tech. report CS-TR-72-252, Department of Computer Science, Stanford University, Stanford, CA (January 1972)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Applied Mathematics,Complejo Tecnológico, University of Málaga, Campus de Teatinos s/n, 29071, Málaga, Spain

    Pablo Guerrero-García & Ángel Santos-Palomo

Authors
  1. Pablo Guerrero-García
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  2. Ángel Santos-Palomo
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Editor information

Editors and Affiliations

  1. Computer Science Department, University of Tennessee, 37996-3450, Knoxville, TN, USA

    Jack Dongarra

  2. Department of Informatics and Mathematical Modelling, Technical University of Denmark, DK-2800, Lyngby, Denmark

    Kaj Madsen

  3. Informatics & Mathematical Modeling, Technical University of Denmark, DK-2800, Lyngby, Denmark

    Jerzy Waśniewski

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© 2006 Springer-Verlag Berlin Heidelberg

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Guerrero-García, P., Santos-Palomo, Á. (2006). A Direct Orthogonal Sparse Static Methodology for a Finite Continuation Hybrid LP Solver. In: Dongarra, J., Madsen, K., Waśniewski, J. (eds) Applied Parallel Computing. State of the Art in Scientific Computing. PARA 2004. Lecture Notes in Computer Science, vol 3732. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11558958_72

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  • DOI: https://doi.org/10.1007/11558958_72

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-29067-4

  • Online ISBN: 978-3-540-33498-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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