Can the solution of a linear program be irrational?

Question

Let's consider a generic linear programming problem. Is it possible that the decision variables of the objective function assume (at the optimal solution) irrational values?

Also, is it possible that some entries of the $A$ matrix are irrational?

Answer 1

Yes.

$$\min x$$ subject to $$\sqrt{2}x=1$$ is a valid linear programming instance.

Answer 2

If the problem is described with rational data, there is always a rational optimal solution. I don't have any reference immediately, but it is a standard result. Search on rational data linear program, polynomial complexity etc, and you will find a lot of material.

Edit: I see my answer was a bit unclear. If the solution to the rational LP is unique, it is rational. If it is non-unique, you can always generate an irrational solution by taking a linear combination of two rational solutions $\alpha x_1 + (1-\alpha)x_2$ where $\alpha$ is an irrational number between $0$ and $1$.