Q:

Prove that Dual linear program does not have finite optimal solution

Consider the following 
$\displaystyle \max z=x+2y\\s.t.-x+y\le-2\\4x+y\le4\\ x,y\ge0$
Find the dual program and prove graphically that D has no finite optimal solution.
Solution
The dual is given by 
$\displaystyle \max -2x+4y\\s.t.-x+4y\ge1\\x+y\ge2\\ x,y\ge0$
In the plot we see that there is no bounded feasible region. And as we have to maximize the objective function $-2x+4y$ then there won't be a fesible solution, because $z$ goes to $\infty$.

Is my solution correct? If it is, is it well justified?

A:

The graph looks ok, but the real trouble is the dual maximizing. Your teacher is right, 

it must be the change from min to max and vice versa in dual problems. 

It is though the same reasoning for the minimum in the dual problem as it is possible to move the level set for $z=-2x+4y$ to $-\infty$ within the feasible region.

