I have a system of inequalities of the form:
$$ X \cdot v_i \leq 0 $$
For some number of vectors $v_i$.
I need to determine if there exists at least one $X$ that can satisfy all of them at once (I don't need to find $X$ I just need to know if it exists).
I need to be able to do this computationally, so graphing the solution won't really work.
$X$ is a vector of the same dimension as $v_i$ for all $i$.
Let $X=\matrix(x_1,...,x_n)\in \mathbb{R}^+\times ... \times \mathbb{R}^+$ denote your variables and $V=\matrix(v_1,...,v_n)$ denote the parameters.
You want to know if the following system has a solution: \begin{align} \sum_{i=1}^nx_iv_i &\le 0 \tag{1}\\ \end{align}
You can feed this to any linear solver (see for example PuLP) with a dummy objective function (e.g. $0$), and the solver will either find a solution, or tell you that the system is not feasible. So calling the solver will give you your certificate.
If your variables are not restricted to non negative values, perform the change of variables ($x_i=x_i'-x_i{''}$) to have this restriction, as explained for example here.
If you want at least one non negative element, then add the constraint $$ \sum_{i=1}^n x_i \ge \epsilon $$ where $\epsilon$ is your tolerance value.