An LP is simultaneously primal infeasible and dual infeasible when the primal constraints are contradictory, leaving no feasible region, while the primal objective direction is unbounded, causing the dual constraints to also be contradictory.

An LP can be both primal infeasible and dual infeasible because the weak/strong duality relationships only guarantee a connection when feasible solutions exist. If the primal has no feasible solution, the dual may also have no feasible solution, especially when the constraints and objective direction create conflicting requirements. In this case, neither problem has a valid optimal solution, so there is no contradiction in both being infeasible.