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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Björck, Å., Duff, I.S.: A direct method for the solution of sparse linear least squares problems. Linear Algebra Appl. 34, 43–67 (1980)
Edlund, O.: A software package for sparse orthogonal factorization and updating. ACM Trans. Math. Software 28(4), 448–482 (2002)
Fletcher, R., Powell, M.J.D.: On themodification of LDLT factorizations. Math. Comp. 29, 1067–1087 (1974)
Alan George, J., Heath, M.T.: Solution of sparse linear least squares problems using Givens rotations. Linear Algebra Appl. 34, 69–83 (1980)
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)
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)
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)
Madsen, K., Nielsen, H.B., Pinar, M.Ç.: A new finite continuation algorithm for linear programming. SIAM J. Optim. 6(3), 600–616 (1996)
Maros, I.: Computational Techniques of the Simplex Method. Kluwer Academic Publishers, Dordrecht (2003)
Matstoms, P.: Sparse linear least squares problems in optimization. Comput. Optim. Applics. 7(1), 89–110 (1997)
Megiddo, N.: On finding primal- and dual-optimal bases. ORSA J. Computing 3(1), 63–65 (1991)
Murty, K.G.: Linear Programming. John Wiley and Sons, Chichester (1983)
Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann (1988)
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)
Nielsen, H.B.: Computing a minimizer of a piecewise quadratic. Tech. report, Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark (September 1998)
Nielsen, H.B.: Algorithms for linear optimization, 2nd edition. Tech. report, Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark (February 1999)
Pan, P.-Q.: A basis-deficiency-allowing variation of the simplexmethod for linear programming. Computers Math. Applic. 36(3), 33–53 (1998)
Santos-Palomo, Á., Guerrero-García, P.: A non-simplex active-setmethod for linear programs in standard form. Stud. Inform. Control 14(2) (June 2005)
Santos-Palomo, Á., Guerrero-García, P.: Updating and downdating an upper trapezoidal sparse orthogonal factorization. Accepted for publication in IMA J. Numer. Anal.
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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)