I am working on a LP problem where I need to find a parameter that creates this optimal solution, given the optimal solution. The problem looks like the following:
$\min x_1 + ax_2$
st
$x_1 + x_2 \geqslant a $
$x_1 - x_2 \leqslant -1 $
My initial thought was to set it up like three equations with three unknowns, however, as I started solving it, it did not make sense. Can anyone help or guide me in the right direction? Much appreciated!
On
(In my answer I changed $x_1$ to $x$ and $x_2$ to $y$)
Note that the set of feasible solution of your LP has only one vertex. Hence, if there is an optimal solution of the problem, then the vertex $(x_0,y_0)$ is an optimal solution (see picture). Moreover, for the vertex $(x_0,y_0)$ the equalities $x_0 + y_0 = a$ and $x_0-y_0 = -1$ hold. Hence, $a=2y_0-1$ is a proper choice of the parameter.
(Note that only for the case a=1 and a=-1 there are more than one optimal solutions of the LP, namely the right or left boundary of the feasible set. However, $(x_0,y_0)$ still belongs to these sets and should be the solution a (simplex based) LP solver provides.)
The task is to find $a$, since we already know $P* = 7$