# Minimal inequality

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 $LaTeX: 2x_1 + x_2 \ge 1$ is a valid inequality. Then, if we decrease the first coefficient to obtain $LaTeX: x_1 + x_2 = 1,$ either this is not valid or it dominates the former, rendering it non-minimal.
More generally, suppose $LaTeX: a^Tx \ge b$ is a valid inequality, and we consider $LaTeX: (a',b')$ such that $LaTeX: a' \le t a$ and $LaTeX: b' \ge t b$ for some positive $LaTeX: t$ such that $LaTeX: (a',b') \neq t (a,b).$ If $LaTeX: (a')^T x \ge b'$ is a valid inequality, it dominates the original one because
For example, $LaTeX: 4x_1 + 2x_2 \ge 3$ dominates $LaTeX: 2x_1 + x_2 \ge 1$ (use $LaTeX: t = 2$), so if this is valid, $LaTeX: 2x_1 + x_2 \ge 1$ cannot be minimal. Every facet-defining inequality is minimal, but not conversely.