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Problem [P]:Minimize $c^Tx$ subject to $Ax ≥ b$, $x ≥ 0$, where $c ≥ 0$. Show that this problem has a finite optimal solution if and only if $b^T\hat y ≤ 0$, where $\hat y$ is the optimal solution to the following problem [D]: maximize $b^Ty$ subject to $A^Ty ≤ 0$, $0 ≤ y ≤ 1$, $1 = [1, 1, . . . , 1]^T$.

I'm stuck on this problem. I could prove that if $b^T\hat y\leq 0$, the problem $P$ has a finite solution. However, I have no idea how to prove the other direction.

Here is my argument:

Let $ b^T y \leq 0$, then since the minimal solution of the problem D and the maximize solution of P are the same, and so $c^T x = b^T y $, hence $c^T x\leq 0$. on the other hand, by the assumption since $x \geq 0$, and $c\geq 0$, then $c^T x =0$. This means that $x=0$. So the problem P has finite solution. I appreciate any help.

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    Can you define all your variables here? – 2017-02-24
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    Your Linear Problem has always Optimal solution (if it is feasible)regardless what you wrote about duality!! Since the vector C has non-negative components taking into account the constraint x>=0 which implies the boundlessness of objective function on feasible solution thus it has optimal solution – 2017-02-27

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I'm not sure if I remember specific proofs for this duality thing properly, but I remember making a table for both the primal and the dual under the different cases in which the primal might be in:

Here's what happens to the primal:

                    | Optimality Conditions Met | Optimality Conditions Not Met
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Solution Feasible   | Optimal                   | Suboptimal
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Solution Infeasible | Superoptimal              | Neither feasible nor super optimal
-------------------------------------------------------------------------------

You know that the dual is optimal when the primal is optimal. But what happens to the dual when the primal is suboptimal? When the primal is superoptimal?.... etc. It could be helpful to try and make that table for the dual as well.

Then from there, try to reason about what happens as you iterate through the simplex algorithm and try to see how the basis variables and for the primal and the dual are changed in the iterations. You should see that the 0 variables in the primal are related to the basis variables for the dual and the basis variables for the primal are the 0 variables for the dual.

Wish I could remember more of this stuff, but I hope I somewhat helped.