I have a linear programming problem that I know is either unbounded or the only feasible solution is the 0 vector (depending on input). In fact, it isn't strictly a linear programming problem since I don't have an objective function, just a list of linear inequalities that need to be solved. I'm using trick to determine whether a solution exists. For each variable $x_i$, I run Mathematica's built in
LinearProgramming function twice, once with objective $-x_i$ and once with objective $x_i$. If there exists non-trivial solutions then for at least one of these objective functions Mathematica reports that the linear program is unbounded.
Here's the problem, however, I would like an example feasible solution. I don't care which one. Just give me any feasible solution that isn't the 0 vector. How can I get Mathematica to do this?