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.

Squeezing a Matrix into Half Precision, with an Application to Solving Linear Systems

“…Despite the large literature on scaling such problems, Downloaded 09/11/20 to 44.224.250.200. Redistribution subject to CCBY license no clear conclusions are available on when or how one should scale; see [8] for a recent experimental study. In any case, our use of these scalings is different from that in previous studies, where the aim of scaling has been to reduce a condition number or to speed up the convergence of an iterative method applied to the scaled matrix.…”

Robust flux balance analysis of multiscale biochemical reaction networks

“…For some problems, such as (3), the scaled constraints $D_{r}S{D}_c\overline{v}=0$ may be satisfied accurately by the scaled solution $\overline{v}$, but when the solution is unscaled, $v=D_c\overline{v}$ may violate S v  = 0 significantly. We refer to [6] for a comprehensive study of scaling and its effects on the performance of the simplex method.…”

Identification, Assessment, and Correction of Ill-Conditioning and Numerical Instability in Linear and Integer Programs

“…In other words, the expression (10) gives an upper bound on the maximum absolute change in x B relative to that of b. Similarly, we can get a relative change in x B by applying the Cauchy-Schwarz inequality to equations (5):…”

“…This involves first scaling the constraints so that the maximum element is 1.0, then scaling the columns to also have a maximum element of 1.0. For details on different scaling methods for LPs, see Elble and Sahinidis [10]. After default scaling, c 2 remains unchanged, but c 1 is rescaled to 3/799988y 1 + 5/799988y 2 + z ≤ 800000/799988.…”