Parametric linear programming and anti-cycling pivoting rules

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. The primal simplex method for minimization problems permits an entering variable at each iteration to be any variable with a negative reduced cost and permits the exiting variable to be any variable that satisfies the minimum ratio rule. As is well-known, any implementation of the procedure is guaranteed to converge if the problem is nondegenerate. In addition, there are two well-known methods for resolving degeneracy. The first of these, the perturbation (or equivalently, the lexicographic) method, avoids cycling by refining the selection rule for the exiting variable (Charnes [1952], Dantzig [1951], Wolfe [1963]). The second method, the combinatorial rule, developed by Bland [1979], avoids cycling by refining the selection rule for both the exiting and entering variables. The situation raises the following natural question: Is there a simplex pivoting procedure for avoiding cycling that does not restrict the minimum ratio rule choice of exiting variables? In this note, we answer this question affirmatively by describing an anti-cycling rule based on a "homotopy principle" that avoids cycling by refining the selection rule for only the entering variable. We also describe an analogous dual pivoting procedure that avoids cycling by refining only the choice of exiting variables. Our procedures are based upon a few elementary observations concerning parametric simplex methods. These observations may be of some importance in their own right, since they may shed light on some theoretical issues encountered in several recent analyses