Certifying the Global Optimality of Quartic Minimization over the Sphere

The quartic minimization over the sphere is an NP-hard problem in the general case. There exist various methods for computing an approximate solution for any given instance. In practice, it is quite often that a global optimal solution was found but without a certification. We will present in this article two classes of methods which are able to certify the global optimality, i.e., algebraic methods and semidefinite program (SDP) relaxation methods. Several advances on these topics are summarized, accompanied with some emerged new results. We want to emphasize that for medium- or large-scaled instances, the problem is still a challenging one, due to an apparent limitation on the current force for solving SDP problems and the intrinsic one on the approximation techniques for the problem.