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

Technical Report MS 96-010

J.A.J. Hall and K.I.M. McKinnon

This paper introduces a class of linear programming examples which cause the simplex method to cycle indefinitely and which are the simplest possible examples which show 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.

Key words: Linear programming, simplex method, degeneracy, cycling.

PDF 96010.pdf
History: Appears with minor modifications in the Southampton e-print archive as math.OC/0012242
Mathematical Programming 100(1), 133-150, 2004.
Note: An important member of the class of examples is illustrated by the following LP Explorer applet