Inferring an LP cost vector from its solution
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&gt;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.