If the dual has an unbounded solution, then its corresponding primal has

Concept. For the standard primal–dual pair — primal: maximise cTx subject to Ax ≤ b, x ≥ 0; dual: minimise bTy subject to ATy ≥ c, y ≥ 0 — the weak duality theorem states that cTx ≤ bTy for every primal-feasible x and every dual-feasible y. Each feasible point of one problem therefore pins a finite bound on the other problem's objective, and one corollary follows immediately: whenever one problem's objective runs off without limit, the other problem cannot possess even a single feasible point.

Application. Read that corollary in the direction this question asks, with the dual as the unbounded problem.

1. The dual is unbounded: being a minimisation, its objective bTy is driven below every finite number as y ranges over the dual feasible set.

2. Suppose, for contradiction, that the primal owned at least one feasible point x0, that is Ax0 ≤ b with x0 ≥ 0.

3. Weak duality applied to that x0 against an arbitrary dual-feasible y gives cTx0 ≤ bTy. The left-hand side is one fixed number, so cTx0 is a lower bound on bTy across the entire dual feasible set.

4. A quantity bounded below cannot be driven below every finite number, which contradicts step 1.

5. The assumption made in step 2 must therefore fail: the primal constraint set {x : Ax ≤ b, x ≥ 0} is empty, so the primal has no feasible solution at all.

Cross-check. Test the result from three independent directions.

• Symmetry: the same weak-duality argument with the two roles swapped shows that an unbounded primal forces an infeasible dual, so unboundedness on one side and an empty constraint set on the other always travel together.

• Non-reversibility: the implication does not run backwards. One problem being infeasible does not force its partner to be unbounded — both members of the pair can be infeasible at the same time. Unboundedness is the direction that carries the force.

• Exclusion: strong duality, where both problems attain equal finite optima, requires both of them to be feasible and bounded, so an unbounded dual removes that case from consideration.

Outcome grid. The complete set of pairings permitted by duality theory: