Proximal Distance Algorithm for Nonconvex QCQP with Beamforming Applications

This paper studies nonconvex quadratically constrained quadratic program (QCQP), which is known to be NP-hard in general. In the past decades, various approximate approaches have been developed to tackle the QCQP, including semidefinite relaxation (SDR), successive convex approximation (SCA), the variable splitting approach, to name a few. While these approaches are effective under some circumstances, they have to either lift the variable dimension or require a feasible starting point, thereby not suitable for the large-scale QC-QP or lack of a feasible starting point. In light of this, this work aims at developing an efficient approach to the QCQP without the above mentioned drawbacks. The crux of our approach is the proximal distance algorithm (PDA), which merges the idea of the penalty method and majorization minimization (MM) to provide an efficient (closed-form) iterative algorithm. To demonstrate the effectiveness of the PDA, we test it on the multicast beamforming applications in wireless communications. Simulation results show that the PDA outperforms state-of-the-art algorithms in terms of delivering a better solution with much less running time.