I have a bunch of vectors $b_i \in R^n$ for $i = 1,\ldots,N$ and a bunch of (indefinite) matrices $A_j$ for $j = 1,\ldots,M$. Let's consider the set $S \subset R^n$ of $x \in R^n$ vectors such that

$$ (x,b_i) \geq 0 $$ $$ (x,A_j x) \geq 0 $$

for all $i$ and $j$. Clearly $0 \in S$ and since both conditions are homogeneous, $\lambda x \in S$ if $x \in S$ for $\lambda \geq 0$. So $S$ contains the origin and is connected. It is either the origin only (trivial case) or a set with positive dimension (non-trivial case).

Are there general theorems for finding out if for given $b_i$ and $A_j$ data the situation is trivial or not?