# Lagrangian relaxation

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?

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