Scaling linear optimization problems prior to application of the simplex method

Abstract

The scaling of linear optimization problems, while poorly understood, is definitely not devoid of techniques. Scaling is the most common preconditioning technique utilized in linear optimization solvers, and is designed to improve the conditioning of the constraint matrix and decrease the computational effort for solution. Most importantly, scaling provides a relative point of reference for absolute tolerances. For instance, absolute tolerances are used in the simplex algorithm to determine when a reduced cost is considered to be nonnegative. Existing techniques for obtaining scaling factors for linear systems are investigated herein. With a focus on the impact of these techniques on the performance of the simplex method, we analyze the results obtained from over half a billion simplex computations with CPLEX, MINOS and GLPK, including the computation of the condition number at every iteration. Some of the scaling techniques studied are computationally more expensive than others. For the Netlib and Kennington problems considered herein, it is found that on average no scaling technique outperforms the simplest technique (equilibration) despite the added complexity and computational cost.