Skip to main content
SpringerLink
Log in

The simplest examples where the simplex method cycles and conditions where expand fails to prevent cycling

  • Published:
Mathematical Programming Submit manuscript

Abstract.

This paper introduces a class of linear programming examples that cause the simplex method to cycle and that are the simplest possible examples showing this behaviour. The structure of examples from this class repeats after two iterations. Cycling is shown to occur for both the most negative reduced cost and steepest-edge column selection criteria. In addition it is shown that the expand anti-cycling procedure of Gill et al. is not guaranteed to prevent cycling.

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. Balinski, M.L., Gomory, R.E.: A mutual primal-dual simplex method. In: R.L. Graves, P. Wolfe, (eds.), Recent Advances in Mathematical programming. McGraw-Hill, New York, 1963

  2. Beale, E.M.L.: Cycling in the dual simplex algorithm. Naval Res. Logistics Quarterly 2, 269–75 (1955)

    MathSciNet  Google Scholar 

  3. Fletcher, R.: Degeneracy in the presence of roundoff errors. Linear Algebra and its Appl. 106, 149–183 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Fletcher, R.: Resolving degeneracy in quadratic programming. Ann. Oper. Res.: degeneracy in optim. problem 47, 307–334 (1993)

    MATH  Google Scholar 

  5. Fletcher R., Hall, J.A.J.: Towards reliable linear programming. In: G.A. Watson and D.F. Griffiths, (eds.), Pitman Research Notes in Mathematics Series 228, Longman Scientific and Technical, 1990, pp. 89–104

  6. Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H.: A practical anti-cycling procedure for linearly constrained optimization. Math. Prog. 45, 437–474 (1989)

    MathSciNet  MATH  Google Scholar 

  7. Goldfarb, D., Reid, J.K.: A practical steepest-edge simplex algorithm. Math. Prog. 12, 361–371 (1977)

    MATH  Google Scholar 

  8. Hall, J.A.J., McKinnon, K.I.M.: Hyper-sparsity in the revised simplex method and how to exploit it. Technical Report MS00-015, Department of Mathematics and Statistics, University of Edinburgh, 2000. Submitted to SIAM Journal of Optimization

  9. Hall, J.A.J., McKinnon, K.I.M.: LP test problems. http://www.maths.ed.ac.uk/hall/PublicLP/, 2002

  10. Harris, P.M.J.: Pivot selection methods of the Devex LP code. Math. Prog. 5, 1–28 (1973); [Reprinted in Math. Prog. Study 2, 30–57 (1975)].

    Google Scholar 

  11. IBM. Optimization Subroutine Library, guide and reference, release 2, 1993

  12. ILOG. CPLEX 6.5 Reference Manual, 1999

  13. Murtagh, B.A., Saunders, M.A.: MINOS 5.4 user’s guide. Technical Report SOL 83-20R, Stanford University, Systems Optimization Laboratory, December 1983 1993. (revised February 1995)

  14. Wolfe, P.: A technique for resolving degeneracy in linear programming. SIAM J. Appl. Math. 11, 205–211 (1963)

    MATH  Google Scholar 

  15. Wunderling, R., Bley, A., Pfender, T., Koch, T.: SOPLEX 1.2.0, 2002

Download references

Author information

Authors and Affiliations

  1. Dept. of Mathematics and Statistics, University of Edinburgh, EH9 3JZ, UK

    J.A.J. Hall & K.I.M. McKinnon

Authors
  1. J.A.J. Hall
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K.I.M. McKinnon
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J.A.J. Hall.

Additional information

Work supported by EPSRC grant GR/J0842

This paper is dedicated to Roger Fletcher, a friend and inspiration to us both. The discovery of Roger’s book, ‘‘Practical Methods of Optimization’’, whilst working in industry, set the first author on the road to Dundee and a career in optimization. Happy 65th birthday, Roger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hall, J., McKinnon, K. The simplest examples where the simplex method cycles and conditions where expand fails to prevent cycling. Math. Program., Ser. B 100, 133–150 (2004). https://doi.org/10.1007/s10107-003-0488-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0488-1

Search

Navigation