Skip to main content

Numerical study of projective methods for linear programming

  • Conference paper
  • First Online:
Optimization

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1405))

Abstract

This paper describes some computational experience with projective methods for linear programming. Rather than compare a particular implementation of a projective algorithm with versions of the simplex method, our purpose is to identify significant characteristics and limitations of a specific class of projective methods. A number of variations are possible within this class, and this study includes a comparison of single-phase and two-phase versions.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 34.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 46.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Anstreicher, K.M., A combined ‘phase I — phase II’ projective algorithm for linear programming, manuscript, Yale School of Organization and Management (1986), to appear in Mathematical Programming.

    Google Scholar 

  2. Ghellinck, G. de, Vial, J.-P., A polynomial Newton method for linear programming, Algorithmica 1, (1986), 425–453.

    Article  MathSciNet  MATH  Google Scholar 

  3. Ghellinck, G. de, Vial, J.-P., An extension of Karmarkar's algorithm for solving a system of linear homogeneous equations on the simplex, Mathematical Programming 39 (1987) 79–92.

    Article  MathSciNet  MATH  Google Scholar 

  4. Chu, E., George, A., Liu, J. and Ng, E., SPARSPAK: Waterloo Sparse Matrix Package, User's Guide for SPARSPAK-A, Research Report CS-84-36, Department of Computer Science, University of Waterloo (November 1984).

    Google Scholar 

  5. Dongarra, J.J., Bunch, J.R., Moler, C.B., Stewart, G.W., LIMPACK User's Guide, SIAM (1979).

    Google Scholar 

  6. Dongarra, J.J., Grosse, E., Distribution of mathematical software via electronic mail, Communications of the ACM, Vol. 30, No. 5 403–407.

    Google Scholar 

  7. Duff, I.S., Reid, J.K., MA-27-A set Fortran subroutines for solving sparse symmetric sets of linear equations, Report AERE R-10533, Computer Science and Systems Division, AERE Harwell, U.K. (1982).

    Google Scholar 

  8. Eisenstat, S.C., Gursky, M.C., Schultz, M.H., Sherman, A.H., Yale sparse matrix package I: The symmetric codes, International Journal of Numerical methods in Engineering 18 (1982), 1145–1151.

    Article  MATH  Google Scholar 

  9. George, A., Ng, E., SPARSPAK: Waterloo sparse matrix package, User's Guide for SPARSPAK-B, Research Report CS-84-37, Department of Computer Science, University of Waterloo (November 1984).

    Google Scholar 

  10. Gill, P.E., Murray, W., Saunders, M.A., Wright M.H., Two steplength algorithms for numerical optimization, Technical report SOL 79-25, Systems optimization Laboratory, department of Operations Research, Stanford University (1979).

    Google Scholar 

  11. Gill, P.E., Murray, W., Wright M.H., Practical optimization, Academic Press (1981).

    Google Scholar 

  12. Gill, P.E., Murray, W., Saunders, M.A., Wright M.H., Users guide for NPSOL (Version 4.0): A Fortran package for nonlinear programming, technical report SOL 86-2, Systems Optimization Laboratory, Department of Operations Research, Stanford University (January 1986; revised Version 4.2 in 1987).

    Google Scholar 

  13. Gill, P. E., Murray, W., Saunders, M.A., Tomlin, J.A., Wright M.H., On projected Newton barrier methods for linear programming and an equivalence to Karmarkar's projective method, Mathematical Programming 36 (1986) 183–209.

    Article  MathSciNet  MATH  Google Scholar 

  14. Gill, P.E., Murray, W., Saunders, M.A., Wright M.H., A note on nonlinear approaches to linear programming, Technical Report SOL 86-7, Systems Optimization Laboratory, Department of Operations Research, Stanford University (April 1986).

    Google Scholar 

  15. Karmarkar, N., A new polynomial-time algorithm for linear programming, Combinatorica 4 (1984) 373–395.

    Article  MathSciNet  MATH  Google Scholar 

  16. Subroutine F04QAF, NAG Fortran Library Manual, Mark 12, Volume 5, Numerical Algorithms Group, Downers Grove, IL U.S.A., and Oxford, U.K. (March 1987).

    Google Scholar 

  17. Paige, C.C., Saunders, M.A., LSQR: An algorithm for sparse linear equations and sparse least-squares, ACM Transactions on Mathematical Software 8 (1982), 43–71.

    Article  MathSciNet  MATH  Google Scholar 

  18. SMPAK User's Guide Version 1.0, Scientific Computing Associates, U.S.A., (1985).

    Google Scholar 

  19. Todd, M.J., Burrell, B.P., An extension of Karmarkar's algorithm for linear programming using dual variables, Algorithmica 1 (1986) 409–424.

    Article  MathSciNet  MATH  Google Scholar 

  20. Todd, M.J., On Anstreicher's combined phase I-phase II projective algorithm for linear programming, Technical Report No.776, School of Operations Research and Industrial Engineering, Cornell University (1988).

    Google Scholar 

  21. Vial, J.-P., A unified approach to projective algorithms for linear programming, CORE Discussion Paper No 8747, Center for Operations Research and Econometrics, Université Catholique de Louvain (September 1987); revised version appears in these proceedings.

    Google Scholar 

  22. Ye, Y., Kojima, M., Recovering optimal dual solutions in Karmarkar's polynomial algorithm for linear programming, Mathematical Programming 39 (1987) 305–317.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Szymon Dolecki

Rights and permissions

Reprints and permissions

Copyright information

© 1989 Springer Verlag

About this paper

Cite this paper

Fraley, C., Vial, JP. (1989). Numerical study of projective methods for linear programming. In: Dolecki, S. (eds) Optimization. Lecture Notes in Mathematics, vol 1405. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0083584

Download citation

  • DOI: https://doi.org/10.1007/BFb0083584

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-51970-6

  • Online ISBN: 978-3-540-46867-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics