# If a point in a linear programme has equal objective function to a point in its dual linear programme they are both optimal

Last edited: 2026-01-28

# Statement

Lemma

For a linear programme given by $A$, $b$, and $c$ if there is a feasible point $x \in \mathbb{R}^n$ and a feasible point $y \in \mathbb{R}^m$ in the dual linear programme such that

$$ c^Tx = b^Ty$$

then they are both optimal points in their respective linear programmes .

# Proof

This follows from the weak duality theorem as

$$c^Tx' \leq b^Ty'$$

for all feasible $x'$ and $y'$ then by definition of their objective functions $x$ and $y$ are optimal.