The Primal-Dual Correspondence Table

Every primal LP has a corresponding dual LP. When the primal has mixed constraint types ($\leq, =, \geq$) and mixed variable sign restrictions, the primal-dual correspondence table tells us, line by line, the form of the dual.

For a maximization primal with constraints indexed $i = 1, \ldots, m$ and variables indexed $j = 1, \ldots, n,$ the correspondence is:

$$\begin{array}{|c|c|}\hline \textbf{Primal (max)} & \textbf{Dual (min)} \\\hline i\text{-th constraint is } \leq b_i & y_i \geq 0 \\ i\text{-th constraint is } = b_i & y_i \text{ free} \\ i\text{-th constraint is } \geq b_i & y_i \leq 0 \\\hline x_j \geq 0 & j\text{-th constraint is } \geq c_j \\ x_j \text{ free} & j\text{-th constraint is } = c_j \\ x_j \leq 0 & j\text{-th constraint is } \leq c_j \\\hline\end{array}$$

The top half maps primal constraints to dual variables. The bottom half maps primal variables to dual constraints.

A helpful pattern: for a max primal in canonical form (all constraints $\leq,$ all variables $\geq 0$), every dual variable is $\geq 0$ and every dual constraint is $\geq.$ Any deviation from canonical on the primal side relaxes (to free/equality) or reverses (to the opposite sign) the corresponding dual restriction.