Say I have the following linear programming formulation in standard form:\begin{equation*}\begin{array}{rl}    \mathbf{x}^* = \underset{\mathbf{x}}{\text{arg}\;\text{min}}        & \mathbf{c}^T\mathbf{x} \\    \mbox{s.t.} & \mathbf{A}\mathbf{x} = \mathbf{b} \\                & \mathbf{x} \ge 0,    \end{array}\end{equation*}with $\mathbf{x}^*$, $\mathbf{A}$ and $\mathbf{b}$ known, but $\mathbf{c}$ unknown.Are there any known methods for computing $\mathbf{c}$ ? 
Inferring an LP cost vector from its solution
ds.algorithms;optimization;linear programming
I guess that you want the set of optimal solutions of the LP as small as possible.  (Otherwise, c=0 certainly makes x* one of the optimal solutions but probably that is not what you want.)One way to obtain such c is as follows.  If x*i=0, then let ci be any strictly negative number.  If x*i>0, then let ci=0.If you want to know all the vectors c with the required property, you can add any linear combination of rows of matrix A to cT.  It is not hard to prove that no other vectors c satisfy the condition.Edit: Entirely rewritten in revision 2 to make it simpler.