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I have an assignment where I need to solve a small problem using Lagrangian relaxation. $$ \min 3x_1-x_2$$ $$x_1-x_2 \ge -1$$ $$-x_1+2x_2\le 5 $$ $$3x_1+2x_2 \ge 3$$ $$6x_1+x_2 \le 15$$ $$x_1,x_2 \ge 0$$ $$x_1,x_2 \in \mathbb{Z}$$

I have formulated the Lagrangian relaxation as: $$3x_1-x_2+v_1(-1-x_1+x_2)+v_2(5+x_1-2x_2)+v_3(3-3x_1-x_2)+v_4(15-6x_1-x_2)\\$$

Initially I have set all V to zero and want to achieve a lower bound. BUT! The problem is unbounded? An initially solution for Lower bound would be infinity for variable $x_2$ since that is the best value that minimizes the objective function if $v = 0$?

To set $x_2$ to infinity since like a bad idea and I wonder how I shall continue?

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Hint.

The feasible set in light blue, looks bounded!

Now in red the integer lattice, in black the minimum $3x_1-x_2$ which is equal to $1$

enter image description here

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  • $\begingroup$ Hi! Thank you for your answer. I thought that the constraints no longer was valid after the relaxation have been done, but I need to check the constraints in the original problem also when solving the langrage-sub problem? $\endgroup$ – Ulrich_Peters Oct 24 '18 at 10:06
  • $\begingroup$ @Ulrich_Peters Please. See the new plot. $\endgroup$ – Cesareo Oct 24 '18 at 10:22

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