Avoid hiding large coefficients

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:

$$\begin{array}{rcl} x - 10^{6} y &\geq& 0 \\ y&\in&[0,10] \end{array}$$

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:

$$\begin{array}{rcl} x - 10 y_1 &\geq& 0\\ y_1 - 10 y_2 &=& 0\\ y_2 - 10 y_3 &=... ...y_4 &=& 0\\ y_4 - 10 y_5 &=& 0\\ y_5 - 10 y &=& 0\\ y&\in&[0,10] \end{array}$$

This form certainly has nicer values in the matrix. However, the solution $$y=-10^{-6},\ x=-1$$ might still be considered feasible as the bounds on variables and constraints might be violated within the tolerances. A better alternative is to reformulate

$$\begin{array}{rcl} x - 10^{6} y &\geq& 0 \\ y&\in&[0,10] \end{array}$$

$$\begin{array}{rcl} x - 10^{3} y' &\geq& 0 \\ y'&\in&[0,10^4]\\ \end{array}$$

where $$10^{-3} y' = y$$ . In this setting, the most negative values for $x$

which might be considered feasible would be $$-10^{-3}$$ , and for the original $y$

variable it would be $$-10^{-9}$$ , which is a clear improvement over the original situation.