primal problem is:
$$\min z = 4x_1-3x_2+5x_3$$
$$7x_1+6x_2+24x_3\le16$$
$$2x_1+5z_2+3x_3\le10$$
$$x_i\ge0$$
the optimal solution is: $(0,2,0), z = -6$
The dual problem is :
$$ \max g = 16w_1+10w_2$$
$$7w_1+2w_2\le4$$
$$6w_1+5w_2\le-3$$
$$24w_1+3w_2\le5$$
$$w_1,w_2\le0$$
I get the optimal solution $g=0$ which is wrong because of the duality theorem, $z(opt)=g(opt)$.
What's wrong with it?
Thanks.
Best Answer
The linear program you give as the dual is correct. However, the optimal solution isn't $g=0$, but rather $g=-6$ at $(w_1,w_2)=\left(0,-\frac{3}{5}\right)$.
Notice that $g=0$ isn't a possibility because if $g=0$ then we have $w_1=w_2=0$ which then does not satisfy the constraint $$6w_1+5w_2\le-3$$ You can also notice that this is the only nontrivial constraint in the dual program - the other constraints are satisfied merely by the $w_1,w_2\le 0$ requirement.