Linear Programming and Standard Form

Question

In order to find the dual of a primal linear program, do I always have to convert it to the standard form first?

For example, if I have the following LP, would the dual also be a min since the LP in standard form is a maximization?

Answer 1

If the primal linear program is a maximization problem, then the dual linear program is a minimization problem and visa-versa.

Objective function

$\text{max} \ 6y_1+10y_2-3y_3$

Restrictions

1. $8y_1-y_3 \leq 5$

The sign $\leq$ due to $x_1 \geq 0$.

2. $y_1+y_2+y_4=-7$

The sign $=$ due to $x_2$ is unconstrained.

3. $-y_1+3y_3-5y_4\leq 2$

The sign $\leq$ due to $x_3 \geq 0$.

4. $-3y_1+y_2+y_3-2y_4\leq 1$

The sign $\leq$ due to $x_4 \geq 0$.

Variables

• $y_1,y_3 \leq 0$ because of the signs, $\leq$, at the first and third constraints at the primal problem.
• $y_2 \geq 0$ because of the sign, $\geq$, at the second constraint at the primal problem.
• $y_4$ is unconstrained because of the equalitiy sign at the fourth constraint at the primal problem.