The positive edge pricing rule for the dual simplex

Abstract

In this paper, we develop the two-dimensional positive edge criterion for the dual simplex. This work extends a similar pricing rule implemented by Towhidi et al. (2014) [24] to reduce the negative effects of degeneracy in the primal simplex. In the dual simplex, degeneracy occurs when nonbasic variables have a zero reduced cost, and it may lead to pivots that do not improve the objective value. We analyze dual degeneracy to characterize a particular set of dual compatible variables such that if any of them is selected to leave the basis the pivot will be nondegenerate. The dual positive edge rule can be used to modify any pivot selection rule so as to prioritize compatible variables. The expected effect is to reduce the number of pivots during the solution of degenerate problems with the dual simplex. For the experiments, we implement the positive edge rule within the dual simplex of the COIN-OR LP solver, and combine it with both the dual Dantzig and the dual steepest edge criteria. We test our implementation on 62 instances from four well-known benchmarks for linear programming. The results show that the dual positive edge rule significantly improves on the classical pricing rules.