I want to solve a linear program using the simplex method, and I know that all my inequalities will pass through the origin (therefore, either my initial solution of (0, ... , 0) is optimal, or the program is unbounded; I'm only really interested in distinguishing between these two cases).
So here's how I think the algorithm would work: Step one is to pick a pivot variable. Step two - and here's where I think things break down - is to find the equation with the smallest value of the constant to the coefficient of the pivot variable. But since my inequalities all pass through the origin, the constant is 0 every time, so all equations are an equally valid choice for the pivot. This reduces the Simplex Method to a brute-force search of the set of basic variables, which would make it run in above-polynomial time.
Is this correct? Or am I missing some feature of the Simplex Method that handles this case?
Thanks in advance!