Cuts and semidefinite liftings for the complex cut polytope

We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices \(xx^\textrm{H}\) $x{x}^{\text{H}}$ , where the elements of \(x \in \mathbb {C}^n\) $x\in {\mathbb{C}}^n$ are mth unit roots. These polytopes have applications in MAX-3-CUT, digital communication technology, angular synchronization and more generally, complex quadratic programming. For \(m=2\) $m=2$ , 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 \(m>2\) $m>2$ and provide a framework to compare them. Further, we consider a second semidefinite lifting of the complex cut polytope for \(m=\infty \) $m=\infty$ . 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.