This sounds like a famous and straightforward question, but I do not know how exactly to solve it, although I have some rather half-baked ideas. I have already looked at these two answers, this and this, which are nice, but do not exactly answer my question.
I have a linear system of inequalities that I can probably rearrange to $ \mathbf{Ax} > \mathbf{b} $, and I am only interested to know the feasibility of the system, that is, yes or no.
In my particular problem, $ \mathbf{A} $ is a square matrix of reals ($ 8 \times 8 $). And there are no sign constraints on the elements of the vector $ \mathbf{x} $. And $ \mathbf{b} = \mathbf{0} $.
I am interested to know how exactly I need to reformulate the problem to be able to feed it to a Linear Programming solver to check for the feasibility of the system.
The issue with LP solvers is that they need an objective function. I know from the answers linked above that this feasibility problem can be converted to an optimization problem by replacing, for instance, the first element of $ \mathbf{b} $ (which is 0) by $ b_1 $, and then maximizing $ b_1 $ under the same constraints.
Now I have an objective function. If a nonnegative $ b_1$ was found, that is, $ b_1 \geqslant 0 $, then the system is feasible. Otherwise, it is not.
Any help, comment, or reference work about my analysis would be appreciated. Look at my specific problem.
Farkas' Lemma gives a criterion for the feasibility of the system $A\mathbf{x} = \mathbf{b}$, $\mathbf{x} \geq \mathbf{0}$, which is an equivalent problem to yours by adding slack variables. It states that such a solution exists if and only if the linear program \begin{align*} min. \, &\mathbf{b}^T\mathbf{y} \\ subject\,to\, &A^T\mathbf{y} \geq \mathbf{0}\end{align*} has a nonnegative optimum. See https://en.wikipedia.org/wiki/Farkas%27_lemma.