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.
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References
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
Beale, E.M.L.: Cycling in the dual simplex algorithm. Naval Res. Logistics Quarterly 2, 269–75 (1955)
Fletcher, R.: Degeneracy in the presence of roundoff errors. Linear Algebra and its Appl. 106, 149–183 (1988)
Fletcher, R.: Resolving degeneracy in quadratic programming. Ann. Oper. Res.: degeneracy in optim. problem 47, 307–334 (1993)
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
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)
Goldfarb, D., Reid, J.K.: A practical steepest-edge simplex algorithm. Math. Prog. 12, 361–371 (1977)
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
Hall, J.A.J., McKinnon, K.I.M.: LP test problems. http://www.maths.ed.ac.uk/hall/PublicLP/, 2002
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)].
IBM. Optimization Subroutine Library, guide and reference, release 2, 1993
ILOG. CPLEX 6.5 Reference Manual, 1999
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)
Wolfe, P.: A technique for resolving degeneracy in linear programming. SIAM J. Appl. Math. 11, 205–211 (1963)
Wunderling, R., Bley, A., Pfender, T., Koch, T.: SOPLEX 1.2.0, 2002
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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.
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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
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DOI: https://doi.org/10.1007/s10107-003-0488-1