I am looking for efficient computational methods for solving a class of linear programs whose right hand side is zero:
$$ \min c^T x \qquad\text{ subject to }\qquad Ax\ge 0 $$
What is the best general purpose algorithm for solving such linear program?
Note 1: $x=0$ is always a basic feasible solution, and moreover, it is the only basic feasible solution. Consequently, the solution to this linear program is either zero or unbounded, and I'm interested in deciding which is the case.
Note 2: The standard simplex algorithm with the minimum ratio rule does not provide a good way of selecting a pivot row because all ratios are zero, so any row is admissible for pivoting (provided the entry is positive).