Cuts and semidefinite liftings for the complex cut polytope - Sinjorgo, Lennart; Sotirov, Renata; Anjos, Miguel F. (Mathematical programming, 2024)
We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices $$xx^H$$ x x H , where the elements of $$x \in \mathbb {C}^n$$ x ∈ C n are m th 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 $$m=∞ $$ 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.