Parametric Linear Programming and Anti-Cycling Pivoting Rules

Abstract

The traditional perturbution (or lexicographic) methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and, therefore,restrict the choice of exiting basic variables. Bland's combinatorial pivoting rule also restricts the choice of exiting variables. Using ideas from parametric linear programming, we develop anti-cycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule. That is, any variable that ties for the ratio rule can leave the basis. A similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables.