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Gurobi tolerances and the limitations of double-precision arithmetic

The default values for these primal and dual feasibility tolerances are <span>$</span>10^{-6}<span>$</span>, and the default for the integrality tolerance is <span>$</span>10^{-5}<span>$</span>. If you choose the range for your inequalities and variables correctly, you can typically ignore tolerance issues entirely.

To give an example, if your constraint right-hand side is on the order of <span>$</span>10^3<span>$</span>, then relative numeric errors from computations involving the constraint (if any) are likely to be less than <span>$</span>10^{-9}<span>$</span>, i.e., less than one in a billion. This is usually far more accurate than the accuracy of input data, or even of what can be measured in practice.

However, if you define a variable <span>$</span>x\in[-10^{-6},10^{-6}]<span>$</span>, then relative numeric error may be as big as 50% of the variable range.

If, on the other hand, you have a variable <span>$</span>x\in[-10^{10},10^{10}]<span>$</span>, and you are using default primal feasibility tolerances; then what you are really asking is for the relative numeric error (if any) to be less than <span>$</span>10^{-16}<span>$</span>. However, this is beyond the limits of comparison for double-precision numbers. This implies that you are not allowing any round-off error at all when testing feasible solutions for this particular variable. And although this might sound as a good idea, in fact, it is really bad, as any round-off computation may result in your truly optimal solution being rejected as infeasible.

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