Given matrix $A \in \Bbb R^{m \times n}$, where $m \ll n$, can I check whether $Ax<0$ has a solution $x \in \Bbb R^{n \times 1}$?
The operation $<$ is taken coordinate-wise. I am not sure but I believe my question is equal to checking whether a given $H$-polytope
$$P = \left\{ x ∈ \Bbb R^n \mid a_i^T x \leq 0, 1 \leq i \leq m \right\}$$
is empty or not.
A hack would be to try to solve the linear optimization problem \begin{align} \max_{\epsilon,x}~~& \epsilon \\ \text{s.t.}& ~~Ax-\epsilon \leq 0 \\ & \epsilon \geq 0 \end{align} Note that if a point $x$ exists such that $Ax < 0$ is true, then the above optimization problem has to find that. The trouble is it can be unbounded. But, this should be a starting point.
