TITLE: "All positive solution" to system of linear equations
QUESTION [8 upvotes]: My question is:

Are there any criteria to decide whether a system of linear equations allows a solution where all variables are greater than 0?

Clearly, I could compute the solution space and check if there is a solution that satisfies the condition. But is there any better way?

REPLY [4 votes]: You are asking to solve the linear feasibility problem $Ax=b$ with $x \succ 0$. This is called the "Phase 1" problem in convex optimization because interior point algorithms require a feasible starting point. Without loss of generality, assume the rows of $A$ are scaled so that $b$ is entirely positive. This Phase 1 problem reduces to solving an alternate linear program $Ax+y=b$ with $x+y\succ 0$, where $x=0$ and $y=b$ is a trivial initial solution. Thus determining feasibility requires solving a linear program.