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I already read this post and its answers and I am still not satisfied.

I want to know how to use weak duality to explain why it is not possible for both primal and dual LP to be unbounded.

Here is one way I can explain: Suppose both primal and dual LP are unbounded. Weak duality implies dual LP is infeasible. So, the dual LP is both unbounded and infeasible, which is impossible (right?), so a contradiction.

Is there any examples of LP problems that are both unbounded and infeasible?

John Davies
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  • There is a subtle difference between "dual infeasible" and "unbounded" which might be confusing you. Unbounded implies dual infeasible, but not the other way around. There are problems which are both primal and dual infeasible, but it doesn't mean they are primal and dual unbounded. – Michal Adamaszek Mar 04 '24 at 06:52

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By definition, an unbounded linear programming problem is one that has feasible solutions with arbitrarily large (positive or negative, depending on whether it's a "maximize" or "minimize" problem) values of the objective function. In particular, these are problems that have feasible solutions, so they are not infeasible.

Robert Israel
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