The Complexity of Computing KKT Solutions of Quadratic Programs

It is well known that solving a (non-convex) quadratic program is NP-hard. We show that the problem remains hard even if we are only looking for a Karush-Kuhn-Tucker (KKT) point, instead of a global optimum. Namely, we prove that computing a KKT point of a quadratic polynomial over the domain $[0,1{]}^n$ is complete for the class CLS = PPAD$\cap$PLS.