If I have an integer programming problem with binary decision variables in a quadratic objective function with quadratic constraints, I can solve it using branch and bound in a few different solvers. But what if my objective function is a polynomial of degree 3:
$$U(x,y,z) = ax + by + cz + dxy + eyz+ fxz + gxyz $$
where $a,...,g$ are constants and $x,y,z$ are binary choice variables. Does that extra $gxyz$ term render simplex or IP algorithms that solve the sub-problem of the branch and bound ineffective? If so, is there any intuition for why that is?
Also, what if my my objective function is quadratic but I have a constraint that includes a polynomial of degree 3 such as $xyz=0$? Am I out of luck?
Is there some work-around for solving either of these problems? (Yes, I know there are only 8 possible outcomes, so brute force would be fine here, but I'm looking for answers for larger problems.)