true or false and justify your answer If a linear program is

If a linear program is infeasible, the dual must be unbounded. If a linear program is unbounded, the dual must be infeasible.

Solution

Suppose the primal minimization program is unbounded. This immediately implies that the dual must be infeasible. Similarly, if the dual is unbounded, this immediately implies that the primal must be infeasible. To see this in the first case, let y be any feasible solution to the dual. Since the primal is unbounded, there exists an x such that cx < b^Ty, contradicting the Weak Duality Theorem. Hence, no such y exists. The other argument can be proved identically.