# Minimal inequality

### From Glossary

In integer programming,
a valid inequality is minimal if it not *dominated* by any valid
inequality. Originally, this was limited to not being able to
decrease any coefficient and remain valid. For example, suppose
is a valid inequality. Then, if we decrease the first coefficient
to obtain either this is not
valid or it dominates the former, rendering it non-minimal.

More generally, suppose is a valid inequality, and we consider such that and for some positive such that If is a valid inequality, it dominates the original one because

For example, dominates (use ), so if this is valid, cannot be minimal. Every facet-defining inequality is minimal, but not conversely.