Some Computational Results on MPI Parallel Implementation of Dense Simplex Method

There are two major variants of the Simplex Algorithm: the revised method and the standard, or tableau method. Today, all serious implementations are based on the revised method because it is more efficient for sparse linear programming problems. Moreover, there are a number of applications that lead to dense linear problems so our aim in this paper is to present some computational results on parallel implementation of dense Simplex Method. Our implementation is implemented on a SMP cluster using C programming language and the Message Passing Interface MPI. Preliminary computational results on randomly generated dense linear programs support our results.

Authors:

• El-Said Badr
• Mahmoud Moussa
• Konstantinos Paparrizos
• Nikolaos Samaras
• Angelo Sifaleras

Keywords:

• Linear Programming
• MPI
• Parallel Implementation
• Simplex Algorithm.

References:

[1] G. B. Dantzig, Linear Programming and Extensions. NJ: Princeton,
Princeton University Press, 1963.
[2] K. H. Borgwardt, "Some distribution independent results about the
asymptotic order of the average number of pivot steps in the simplex
method", Mathematics of Operations Research, vol. 7, no. 3, pp. 441-
462, 1982.
[3] I. Maros, and G. Mitra, "Investigating the sparse simplex method on a
distributed memory multiprocessor", Parallel Computing, vol. 26, pp.
151-170, 2000.
[4] D. Klabjan, L. E. Johnson, and L. G. Nemhauser, "A parallel primal-dual
simplex algorithm", Operations Research Letters, vol. 27, no. 2, pp. 47-
55, 2000.
[5] J. Eckstein, I. Bodurglu, L. Polymenakos, and D. Goldfarb, "Data-
Parallel Implementations of Dense Simplex Methods on the Connection
Machine CM-2", ORSA Journal on Computing, vol. 7, no. 4, pp. 402-
416, 1995.
[6] S. P. Bradley, U. M. Fayyad, and O. L. Mangasarian, "Mathematical
Programming for Data Mining: Formulations and Challenges",
INFORMS Journal on Computing, vol. 11, no. 3, pp. 217-238, 1999.
[7] I. Foster, Designing and Building Parallel Programs: Concepts and
Tools for Parallel Software Engineering. Addison-Wesley, 1995.
[8] W. Gropp, E. Lusk, and A. Skjellum, Using MPI: Portable Parallel
Programming with the Message Passing-Interface. MIT Press, 1994.