QCQP With Extra Constant Modulus Constraints: Theory and Application to SINR Constrained Mmwave Hybrid Beamforming

The constant modulus constraint is widely used in analog beamforming, hybrid beamforming, intelligent reflecting surface design, and radar waveform design. The quadratically constrained quadratic programming (QCQP) problem is also widely used in signal processing. However, the QCQP with extra constant modulus constraints was not systematically studied in mathematic programming and signal processing. For example, the multiple quality of service (QoS) constrained analog beamforming is rare, while the QoS constrained digital beamforming methods are abundant. We propose to tackle the QCQP with extra constant modulus constraints problem by solving a series of subproblems with linear programming (LP) under extra constant modulus constraints. Under mild condition, the strong duality between the LP with extra constant modulus constraints and its dual problem is established. Then, by using the optimal solutions from the subproblems, the QCQP with extra constant modulus constraints problem is solved with a monotonically converged algorithm, and all converged solutions are Karush-Kuhn-Tucker (KKT) points. As an application of the proposed method, the signal-to-interference-plus-noise-ratio (SINR) constrained hybrid beamforming is solved. Simulation results show that the transmit power of the proposed method is similar to that of the semidefinite relaxation (SDR) method, while the computational time of proposed method is much smaller than that of the SDR method. The proposed method always provides feasible solution, while the SDR method does not always provide feasible solution.