Making progress during a stall in the simplex algorithm

All the standard methods for avoiding cycling in the simplex algorithm use row selection rules for resolving degeneracy. The method proposed in this paper does not. Instead, the Gass-Saaty parametric method applied to the objective (cost) form is used to choose the incoming column. The pivot-row choice among blocking rows, i.e., those tied for pivot, is arbitrary. A simple anticycling device is used which avoids dual degeneracy of the parametrized objective with “probability one”. Tests were run on nine highly degenerate practical test problems ranging in size from small to large. Using MINOS software, the standard simplex method required 31,195 iterations, of which 20,504 (or 66%) were blocked pivots. Using the parametric scheme required 13,812 iterations, of which 67% were blocked. The reduction in the number of iterations for this set of highly degenerate test problems was 56%. The CPU-time reduction was 48%.