Cuts and semidefinite liftings for the complex cut polytope

We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices xx^H, where the elements of x ∈ℂ^n are mth unit roots. These polytopes have applications in MAX-3-CUT, digital communication technology, angular synchronization and more generally, complex quadratic programming. For m=2, the complex cut polytope corresponds to the well-known cut polytope. We generalize valid cuts for this polytope to cuts for any complex cut polytope with finite m>2 and provide a framework to compare them. Further, we consider a second semidefinite lifting of the complex cut polytope for m=∞. This lifting is proven to be equivalent to other complex Lasserre-type liftings of the same order proposed in the literature, while being of smaller size. We also prove that a second semidefinite lifting of the complex cut polytope for n = m=3 is exact. Our theoretical findings are supported by numerical experiments on various optimization problems.