# What happens to an equality constraint in the primal when you translate to the dual?

\begin{align} \text{maximize} &\quad x_1 +3x_2 & & \\ \text{subject to} &\quad x_1 - x_2 = 2 \\ & -2x_1 +3x_2 \ge 5 \\ & \quad\quad\quad x_2 \ge 0 \end{align}

I was able to find the dual LP, but I am not seeing any connections to the original LP. Can I get some pointers?

Find the dual LP.

I first converted the LP to standard form then introduced a new variable.

Let $$x_1=r_1-r_2$$ \begin{align} \text{minimize} &\quad 2r_1-2r_2 -5x_2 & & \\ \text{subject to} &\quad r1-r_2+2x_2\geq 1 \\ & -r1+r_2-2x_2\geq -1 \\ & -r1+r_2-3x_2\geq 3 \end{align}

What happens to an equality constraint in the primal when you translate to the dual?

What happens to a free variable in the primal when you translate to the dual?

• Have your read the table I´ve posted here? I wonder why you cannot answer the questions by yourself. Commented Feb 19, 2021 at 8:19

• An equality constraint in the primal corresponds to a free variable in the dual formulation.

• A free variable in the primal corresponds to an equality constraint in the dual formulation.

$$\max x_1+3x_2$$

subject to $$x_1-x_2=2$$ $$-2x_1+3x_5 \ge 5$$ $$x_2 \ge 0$$

We can associate a dual variable $$p$$ to the first equation and $$q$$ to the second constraint. $$p$$ is a free variable.

Since $$x_1$$ is a free variable in the primal formulation, the corresponding constraint in the dual variable in an equality which is $$p-2q=1$$.