The Dual of the Dual is the Primal

Every linear program has a dual, computed by applying the primal-dual correspondence table. A natural question follows: what happens if we dualize the dual?

The dual of the dual of any LP is the original primal LP.

This is the involution property of LP duality: dualization is its own inverse.

We can verify this on the symmetric primal-dual pair. Let $(P)$ be the primal

$$(P): \quad \max\; \mathbf{c}^T \mathbf{x} \quad \text{s.t.} \quad A\mathbf{x} \le \mathbf{b}, \quad \mathbf{x} \ge \mathbf{0}.$$

Its dual $(D)$ is

$$(D): \quad \min\; \mathbf{b}^T \mathbf{y} \quad \text{s.t.} \quad A^T \mathbf{y} \ge \mathbf{c}, \quad \mathbf{y} \ge \mathbf{0}.$$

Now apply the correspondence rules to $(D).$ It is a minimization with $\ge$ constraints and $\ge 0$ variables, so its dual is a maximization with $\le$ constraints and $\ge 0$ variables, and the coefficient matrix transposes back from $A^T$ to $A{:}$

$$(D'): \quad \max\; \mathbf{c}^T \mathbf{x} \quad \text{s.t.} \quad A \mathbf{x} \le \mathbf{b}, \quad \mathbf{x} \ge \mathbf{0}.$$

This is exactly $(P).$

Consequence. The primal-dual relationship is symmetric. Neither LP in the pair is intrinsically 'the primal' — either one may be called the primal, and the other is its dual.