Are there any theorems that can be used to judge if an all-integer programming problem has a feasible solution?
I am NOT asking if we have any computational solvers that can be used to find a feasible solution to an all-integer programming problem. I think I am asking a more "abstract" question: do we have any theorems to ensure that if the objective function and constraints satisfy some properties, then we must have a feasible solution to an all-integer programming problem?