My goal is this. I have variables $a_1, a_2, ..., a_n \geq 0$. I also must choose a set of inequalities. My choices are as follows:
$\forall S \subseteq [n]$, choose $\sum \limits _{i \in S} a_i \geq 1$ or $\sum \limits _{i \in S} a_i < 1$.
How can I make a list of the sets of choices that give me a consistent set of inequalities, that is, a set of inequalities with a nonempty feasible region?
For example, if I have $a_1, a_2$ the list of sets of choices would look like this:
$1. a_1 \geq 1, a_2 \geq 1, a_1 + a_2 \geq 1 \\ 2.a_1 \geq 1, a_2 < 1, a_1 + a_2 \geq 1 \\ 3. a_1 < 1, a_2 \geq 1, a_1 + a_2 \geq 1 \\ 4. a_1 < 1, a_2 < 1, a_1+a_2 \geq 1 \\ 5. a_1 < 1, a_2 < 1, a_1 + a_2 < 1 \\ $
because each set of inequalities on this list is consistent.
My list would not include $a_1 \geq 1, a_2 \geq 1, a_1 + a_2 < 1$, for example, because this set of inequalities in inconsistent given the constraints.
It is not as simple as randomly choosing for each subset of $[n]$. For example, if I have $a_1, a_2, a_3, a_4$, I cannot simultaneously choose
$a_1 + a_2 \geq 1$
$a_3 + a_4 \geq 1$
$a_1 + a_3 < 1$
$a_1 + a_4 < 1$
$a_2 + a_4 < 1$
$a_2 + a_3 < 1$
because I know that one of $a_1,a_2$ is at least $1/2$, and also one of $a_3,a_4$ is at least $1/2$.
I have tried to come up with a recursive enumeration, but even for a small number of variables, the number of possibilities seems to get out of control quickly. I've also thought about using the simplex algorithm by putting an artificial variable in each $\geq$ inequality and trying to minimize the sum of the artificial variables, but it's not clear how I could apply this to the general case. I've also thought about enumerating regions using a list of vertices, which I can find by taking sets of inequalities as equations and finding unique intersections, but in certain cases, a feasible region does not contain any of its vertices.
I feel like there should be an easy answer to this question, but for some reason I can't come up with anything. I believe this problem is related to algebraic combinatorics, but I can't find anything in the resources that I have to answer this question. By the way, this is not homework; this is just something related to a problem I came up with in graph theory.
Any help or ideas would be appreciated. Thank you.