An Iterative Method for Nonconvex Quadratically Constrained Quadratic Programs

Abstract: This paper examines the nonconvex quadratically constrained quadratic programming (QCQP) problems using an iterative method. One of the existing approaches for solving nonconvex QCQP problems relaxes the rank one constraint on the unknown matrix into semidefinite constraint to obtain the bound on the optimal value without finding the exact solution. By reconsidering the rank one matrix, an iterative rank minimization (IRM) method is proposed to gradually approach the rank one constraint. Each iteration of IRM is formulated as a convex problem with semidefinite constraints. An augmented Lagrangian method, named extended Uzawa algorithm, is developed to solve the subproblem at each iteration of IRM for improved scalability and computational efficiency. Simulation examples are presented using the proposed method and comparative results obtained from the other methods are provided and discussed.