# How do I find the Dual of the LP

min $$= 4y_1 +2y_2 - y_3$$

$$s.t. y_1 + 2y_2 +2y_3 <= 6$$

$$s.t. y_1- y_2 + 2y_3 = 8$$

$$y_1, y_2 >= 0, y_3$$urs

How do I find the dual to this LP? What confuses me is the = sign. Also the urs variable. I'm not sure how to act on these.

• You can write $y_3=y_4-y_5$ where $y_4,y_5\geqslant 0$. Commented Jun 3, 2019 at 18:17
• I know. But then how do you use this to find the dual?
– user634512
Commented Jun 3, 2019 at 18:23

With the table below you do not need to transform $$y_3$$. You have to read the table from right to left, since the primal problem is min-problem. The dual of the LP is

$$\texttt{max} \ \ 6u_1+8u_2$$

$$u_1+u_2\leq 4 \quad (y_1)$$

$$2u_1-u_2\leq 2 \quad (y_2)$$

$$2u_1+2u_2=8 \quad (y_3)$$

$$u_1\leq 0, u_2 \text{ unrestr.}$$

• isn't a max problem supposed to be with <= constraints? Also, why do we put = to the last constraint and not to any other?
– user634512
Commented Jun 3, 2019 at 19:10
• @ThePoorJew "isn't a max problem supposed to be with <= constraints?" Mostly. But there can be other constraints as well. With the same argument you can ask a similar question for the primal. In general I don´t say that there exists a optimal solution (for both). "Also, why do we put = to the last constraint and not to any other?" Because $y_3$ is unrestricted (seventh line of the table). Commented Jun 3, 2019 at 19:17
• @ThePoorJew I´ve added the corresponding variables. You can try to convert back the dual to the primal. It should be the same primal as you have in your question. Commented Jun 3, 2019 at 19:21
• But, from what I know,if our primal is a nonnormal form LP, I have to multiply the <= constraint with (-1), yielding -y1-2y2-2y3>=-6. And then proceed. Am I wrong?
– user634512
Commented Jun 3, 2019 at 19:25
• @ThePoorJew Usually you solve an LP with the simplex algorithm. In case of $\geq$-constraints you add an artificial constraint. But if you want to multiply a $\geq$ constraint by (-1) to obtain a "normal" LP, you can do it. It doesn´t affect the dual problem. Commented Jun 3, 2019 at 19:44