Wikipedia: Quadratic programming says that a positive definite quadratic programming (QP) with linear constraints can be solved in polynomial time: “For positive definite Q, the ellipsoid method solves the problem in polynomial time.[6]”
On the other hand, mixed integer linear program (MILP) can be converted into quadratically constrained quadratic program (QCQP). We know that MILP is NP-hard so QCQD must be NP-Hard in general (again from: Wikipedia:Quadratically constrained quadratic program).
So does that mean that if you have a quadratic term in your constrains the problem is NP-hard and if it is in objective function and it is positive definite, then it can be solved at polynomial time?
