# Complementary slackness and optimal solution for primal

We have primal, minimize $$z = 3u_1 + 0.5u_2$$

subject to $$u_1 - 2u_2 \leq 4 \\ u_1 + u_2 \leq 2 \\ u_1, u_2 \geq 0$$ I found the dual $$\text{max: } z' = 4v_1 + v_2 \\ \text{subject to: } \\ v_1 + v_2 \leq 3 \\ -2v_1 + v_2 \leq 0.5 \\ v_1, v_2 \geq 0$$ And that the solution is $$(3, 0)$$ for the dual. How do I use complementary slackness to find solutions to the primal.

• Step 1 is - find which variables are paired with which constraints. Step 2 is - find what complementary slackness tells you about each pair. Step 3 is - solve the equations you get. Which step are you stuck at? Mar 27 at 22:13
• Any further questions? Mar 28 at 18:44

$$\text{max: } z' = 4v_1 + v_2 \\ \text{subject to: } \\ v_1 + v_2 \leq 3 \\ -2v_1 + v_2 \leq 0.5 \\ v_1, v_2 \leq 0$$ $$\texttt{Edit}$$: I´ve changed the sign of the inequalities. The optimal solution is $$(v_1^*,v_2^*)=(0,0)$$. Thus the slack variables for the dual are positive and therefore not zero. That means that the corresponding variables of the primal problem are zero. 