Cuts and semidefinite liftings for the complex cut polytope

We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices xxᴴ, where the elements of x ∈ ℂⁿ 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. Our theoretical findings are supported by numerical experiments on various optimization problems.