By the strong duality theorem we know that LP can have 4 possible outcomes:
- dual and primal are both feasible,
- dual is unbounded and primal is infeasible,
- dual is infeasible and primal is unbounded,
- dual & primal are both infeasible.
Given the primal program:
Maximize $z = a x_1 + b x_2$
$c x_1 + d x_2 \leq e$
$f x_1 + g x_2 \leq h$
$x_1, x_2 \geq 0$
We can construct the dual program :
Minimize $w = e y_1 + h y_2$
$c y_1 + f y_2 \geq a$
$d y_1 + g y_2 \geq b$
$y_1, y_2 \geq 0$
And my question is: is it possible to assign values for $a,b,c,d,e,f,g,h$ such that both primal and dual are infeasible?
I tried to came up with values but the case was always that one of them (dual or primal) was infeasible and the other was unbounded.