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Consider the minimization problem

$$ \min 5x_1 + 2x_2 + 4x_3 $$ subject to \begin{align} 3x_1 + x_2 + 2x_3 & \geq 4 \\ 6x_1 + 3x_2 + 5x_3 & \geq 10 \\ x_1, x_2, x_3 & \geq 0 \end{align} My question: I have found the dual problem to the minimazation that being $$ \max 4y_1 + 10y_2 $$ subject to \begin{align} 3y_1 + 6y_2 & \leq 5 \\ y_1 + 3y_2 & \leq 2 \\ 2y_1 + 5y_2 & \leq 4 \\ y_1,y_2 & \geq 0 \end{align} and have found the optimal solution $y* = (y_1*,y_2*) = (1,1/3)$.

However, now I am asked to specify an optimal solution for both the primal and the dual problem but I am not sure how to? I tried to solve the minimization problem using the simplex method, but I just couldn't figure it out. All the videos, I have watched, said to do it by solving the dual problem.. What do I do?

Thanks in advance.

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  • $\begingroup$ You can solve them both by the simplex method. Also, the primal has three decision variables, so your solution is incomplete. $\endgroup$
    – sven svenson
    Oct 19, 2020 at 17:25
  • $\begingroup$ I meant that $y* = (1,1/3)$. Isn't that correct? $\endgroup$
    – Mathias
    Oct 19, 2020 at 17:26
  • $\begingroup$ I think I figured it out.. $\endgroup$
    – Mathias
    Oct 19, 2020 at 17:38
  • $\begingroup$ @Mathias What have you figured out? $\endgroup$
    – callculus42
    Oct 20, 2020 at 14:06

1 Answer 1

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You can apply complementary slackness to obtain $x_3=0$ (because $2y_1^*+5y_2^*<4$) and $3x_1 + x_2 + 2x_3 = 4$ and $6x_1 + 3x_2 + 5x_3 = 10$ (because $y_1^* > 0$ and $y_2^* > 0$).

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