Skip to main content
SpringerLink
Log in

A Dual Projective Pivot Algorithm for Linear Programming

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Recently, a linear programming problem solver, called dual projective simplex method, was proposed (Pan, Computers and Mathematics with Applications, vol. 35, no. 6, pp. 119–135, 1998). This algorithm requires a crash procedure to provide an initial (normal or deficient) basis. In this paper, it is recast in a more compact form so that it can get itself started from scratch with any dual (basic or nonbasic) feasible solution. A new dual Phase-1 approach for producing such a solution is proposed. Reported are also computational results obtained with a set of standard NETLIB problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. 1._ E.M.L. Beale, “An alternative method for linear programming,” Proceedings of Cambridge Philosophical Society, vol. 50, pp. 513–523, 1954.

    Google Scholar 

  2. M. Benichou, J.M. Gauthier, G. Hentges, and G. Ribiere, “The efficient solution of linear programming problems—Some algorithmic techniques and computational results,” Mathematical Programming, vol. 13, pp. 280–322, 1977.

    Google Scholar 

  3. A. Bjorck, Numerical Methods for Least Squares Problems, SIAM, Philadelpha, 1996.

    Google Scholar 

  4. R.G. Bland, “New finite pivoting rules for the simplex method,” Mathematics of Operations Research, vol. 2, pp. 103–107, 1977.

    Google Scholar 

  5. A. Charnes, Optimality and degeneracy in linear programming, Econometrica, vol. 2, pp. 160–170, 1951.

    Google Scholar 

  6. G.B. Dantzig, Maximization of a linear function of variables subject to linear inequalities, in Activity Analysis of Production and Allocation, T.C. Koopmans (Ed.), John Wiley & Sons: New York, 1951, pp. 339–347.

    Google Scholar 

  7. G.B. Dantzig, A. Orden, and P. Wolfe, “The generalized simplex method for minimizing a linear form under linear inequality restraints,” Pacific Journal of Mathematics, vol. 5, pp. 183–195, 1955.

    Google Scholar 

  8. J.J.H. Forrest and D. Goldfarb, “Steepest-edge simplex algorithms for linear programming,” Mathematical Programming, vol. 57, pp. 341–374, 1992.

    Google Scholar 

  9. D.M. Gay, “Electronic mail distribution of linear programming test problems,” Mathematical Programming Society COAL Newsletter, vol. 13, pp. 10–12, 1985.

    Google Scholar 

  10. P.E. Gill, S.J. Hammarling, W. Murray, M.A. Saunders, and M.H. Wright, User's Guide for LSSOL (Version 1.0): A Fortran Package for Constrained Linear Least-Squares and Convex Quatratic Programming, Department of Engineeing Economic Systems & Operations Research, Stanford University, 1986.

  11. W. Givens, “Computation of plane unitary rotations transforming a general matrix to triangular form,” SIAM J. App. Math., vol. 6, pp. 26–50, 1958.

    Google Scholar 

  12. G.H. Golub and C.F. Van Loan, Matrix Computations. John Hopkins University Press: Baltimore, MD, 1983.

    Google Scholar 

  13. W.W. Hager, “The LP Dual active set algorithm,” in High Performance Algorithms and Software in Nonlinear Optimization, R. De Leone et al. (Eds.), Kluwer, Dordrecht, 1998, pp. 243–254.

    Google Scholar 

  14. W.W. Hager, “The dual active set algorithm and its application to linear programming,” Computational Optimization and Applications, vol. 21, pp. 263–275, 2002.

    Google Scholar 

  15. W.W. Hager, C.-L. Shih, and E.O. Lundin, Active Set Strategies and the LP Dual Active Set Algorithm, Department of Mathematics, University of Florida, Gainesville, FL, Aug., 1996.

    Google Scholar 

  16. P.M.J. Harris, “Pivot selection methods of the Devex LP code,” Mathematical Programming Study, vol. 4, pp. 30–57, 1975.

    Google Scholar 

  17. B. Hattersley and J. Wilson, “A dual approach to primal degeneracy,” Mathematical Programming, vol. 42, pp. 135–145 1988.

    Google Scholar 

  18. A.J. Hoffman, Cycling in the simplex algorithm, Report No. 2974, Nat. Bur. Standards, Washington D.C., 1953.

    Google Scholar 

  19. P.-Q. Pan, Practical finite pivoting rules for the simplex method, OR Spektrum, vol. 12, pp. 219–225, 1990.

    Google Scholar 

  20. P.-Q. Pan, “A simplex-like method with bisection for linear programming,” Optimization, vol. 22, no. 5, pp. 717–743, 1991.

    Google Scholar 

  21. P.-Q. Pan, “A dual projective simplex method for linear programming,” Computers and Mathematics with Applications, vol. 35, no. 6, pp. 119–135, 1998a.

    Google Scholar 

  22. P.-Q. Pan, “A basis-deficiency-allowing variation of the simplex method,” Computers and Mathematics with Applications, vol. 36, no. 3, pp. 33–53, 1998b.

    Google Scholar 

  23. P.-Q. Pan, “A projective simplex method for linear programmming,” Linear Algebra and Its Applications, vol. 292, pp. 99–125, 1999.

    Google Scholar 

  24. P.-Q. Pan, “A projective simplex algorithm Using LU decomposition,” Computers and Mathematics with Applications, vol. 39, pp. 187–208, 2000.

    Google Scholar 

  25. P.-Q. Pan and Y.-P. Pan, “A phase-1 approach for the generalized simplex algorithm,” Computers and Mathematics with Applications, vol. 42, pp. 1455–1464, 2001.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Southeast University, People's, Republic of China

    Ping-Qi Pan

Authors
  1. Ping-Qi Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pan, PQ. A Dual Projective Pivot Algorithm for Linear Programming. Computational Optimization and Applications 29, 333–346 (2004). https://doi.org/10.1023/B:COAP.0000044185.69640.54

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:COAP.0000044185.69640.54

Search

Navigation