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

## TL;DR

This paper introduces a spatial branch-and-cut method for solving nonconvex QCQPs with bounded complex variables, enhancing relaxations with valid inequalities and specialized branching rules, demonstrated on power flow and quadratic programming problems.

## Contribution

It develops a novel spatial branch-and-cut algorithm incorporating valid inequalities and new branching rules for nonconvex QCQPs with complex variables.

## Key findings

- Effective in solving AC optimal power flow problems.
- Improves relaxation strength through valid inequalities.
- Reduces domain via bound tightening procedures.

## Abstract

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.