As we said before, a typical recommendation for improving numerics is to limit the range of constraint matrix coefficients. The rationale behind this guideline is that terms to be added in a linear expression should be of comparable magnitudes so that rounding errors are minimized. For example:
is usually considered a potential source of numerical instabilities due to the wide range of the coefficients in the constraint. However, it is easy to implement a simple (but useless) alternative:
This form certainly has nicer values in the matrix. However, the solution might still be considered feasible as the bounds on variables and constraints might be violated within the tolerances. A better alternative is to reformulate
where . In this setting, the most negative values for which might be considered feasible would be , and for the original variable it would be , which is a clear improvement over the original situation.