A spatial branch-and-cut method for nonconvex QCQP with bounded complex variables

• Published in: Mathematical programming Vol. 165; no. 2; pp. 549 - 577
• Main Authors: Chen, Chen, Atamtürk, Alper, Oren, Shmuel S.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2017; Springer Nature B.V
• Subjects: Algorithms; Calculus of Variations and Optimal Control; Optimization; Combinatorics; Complex variables; Constraints; Control theory; Experiments; Full Length Paper; Inequalities; Mathematical analysis; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Mathematics of Computing; Matrix methods; Numerical Analysis; Power flow; Quadratic programming; Real variables; Semidefinite programming; Theoretical; Variables; 90C26; 65K05
• ISSN: 0025-5610; 1436-4646
• DOI: 10.1007/s10107-016-1095-2

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 × 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.