  We develop a spatial branch-and-cut approach for nonconvex Quadratically
Constrained Quadratic Programs with bounded complex variables (CQCQP). Linear
valid inequalities are added at each node of the search tree to strengthen
semidefinite programming relaxations of CQCQP. These valid inequalities are
derived from the convex hull description of a nonconvex set of $2 \times 2$
positive semidefinite Hermitian matrices subject to a rank-one constraint. We
propose branching rules based on an alternative to the rank-one constraint that
allows for local measurement of constraint violation. Closed-form bound
tightening procedures are used to reduce the domain of the problem. We apply
the algorithm to solve the Alternating Current Optimal Power Flow problem with
complex variables as well as the Box-constrained Quadratic Programming problem
with real variables.
