How to determine whether a system of linear inequalities has a positive solution or not?
Is there any poly-time algorithm to do this? Or the best algorithms known are no less complex than algorithms for solving set of linear inequalities?
How to determine whether a system of linear inequalities has a positive solution or not?
Is there any poly-time algorithm to do this? Or the best algorithms known are no less complex than algorithms for solving set of linear inequalities?
Your problem is known as Linear Programming (if you change positive to non-negative). Usually linear programming is thought of as an optimization problem, but in fact the optimization problem is equivalent to feasibility, which is exactly what you're asking: whether a system of inequalities has any solution.
If you really want to ask whether there's a positive solution, then what you can do is take your system of inequalities $Ax \geq b$ and add the constraint $x_i \geq m$, maximizing over $m$. If the maximum is $m > 0$, then there is a strictly positive solution, otherwise there isn't.