Question

In: Advanced Math

Prove that if the primal minimisation problem is unbounded then the dual maximisation problem is infeasible.

  1. Prove that if the primal minimisation problem is unbounded then the dual maximisation

    problem is infeasible.

Solutions

Expert Solution

Suppose this is not the case, so that the dual maximization problem

is feasible, but the primal minimization problem

is unbounded.

Fix a feasible solution to the dual problem. Then , so that for all we have

Since the primal problem is unbounded, for every there is such that and . But then, from we get

However, recall that is a fixed vector, and is a fixed feasible solution to the dual. Therefore, is a real constant. Thus, the above shows that for every we have , a constant; this is absurd.

Therefore, the dual problem must be infeasible.

Colby Messinger answered 2 years ago
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