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We develop a spatial branch-and-cut approach for nonconvex QuadraticallyConstrained Quadratic Programs with bounded complex variables (CQCQP). Linearvalid inequalities are added at each node of the search tree to strengthensemidefinite programming relaxations of CQCQP. These valid inequalities arederived from the convex hull description of a nonconvex set of $2 \times 2$positive semidefinite Hermitian matrices subject to a rank-one constraint. Wepropose branching rules based on an alternative to the rank-one constraint thatallows for local measurement of constraint violation. Closed-form boundtightening procedures are used to reduce the domain of the problem. We applythe algorithm to solve the Alternating Current Optimal Power Flow problem withcomplex variables as well as the Box-constrained Quadratic Programming problemwith real variables.

A Spatial Branch-and-Cut Method for Nonconvex QCQP with Bounded Complex Variables