Assume we have a linear optimization problem as below:
$$Min ~~z=c^Tx$$ Subject to: $$A_{eq}x=b_{eq}$$ $$Ax \ge b$$ $$x \ge 0$$
Matrix $A_{eq}$ and $A$ are given, but $c^T,~b_{eq}$ and $b$ are not available. Instead, we do know the actual values of the dual variables associated with the equality constraints. Is there a way to extract matrices $c^T,~b_{eq}$ and $b$ ? If this is not possible, what is the least information I still need to know to be able to extract $c^T,~b_{eq}$ and $b$ ? Would it help if I had the dual variables associated with the inequality constraints too?
I think I should be able to fill out the last snapshot of the simplex table using the information I know about the model, and then from there I might be able to obtain the unknown matrices, but haven't been able to do so.
The optimality conditions are:
$c - A_{eq}^T \lambda_{eq} - A^T \lambda - \mu = 0$
$\lambda^T(Ax - b) = 0$
$\mu^T x = 0$
$\lambda \ge 0$, $\mu \ge 0$
where $\lambda_{eq}$ are the equality multipliers, $\lambda$ the inequality multipliers and $\mu$ the bound multipliers.
So depending on which information you have, you may be able to figure out the other quantities.