An efficient global algorithm for nonconvex complex quadratic problems with applications in wireless communications

In this paper, we consider a class of nonconvex complex quadratic programming (CQP) problems. By using the polar coordinate representations of the complex variables, we first derive a new semidefinite programming (SDP) relaxation for problem (CQP), which is tighter than the conventional SDP relaxation. Based on the newly derived enhanced SDP relaxation, we further propose an efficient branch-and-bound algorithm for solving problem (CQP). Key features of our proposed branch-and-bound algorithm are: (1) it is guaranteed to find the global solution of the problem (within any given error tolerance); (2) it is computationally efficient because it carefully utilizes the special structure of the complex variables, i.e., using their polar coordinate representations. We apply our proposed algorithm to solve the virtual beamforming design problem in the single-hop network and the maximum-likelihood (ML) multi-input multi-output (MIMO) detection problem arising from wireless communications. Simulation results show that our proposed algorithm can efficiently solve these problems.