Fast proximal gradient algorithm for single-group multicast beamforming

Wireless multicasting plays an important role in currrent wireless communication standards (UMTS, LTE, etc) to support massive multimedia services. Wireless multicasting in the form of beamforming is regarded as an effective way to exploit the broadcast nature of wireless transmission to boost network throughput and ensure Quality of Service (QoS). However, the associated max-min fair (MMF) multicast beamforming design is a difficult constrained optimization problem, due to its non-smoothness and nonconvexity. Existing solutions for this NP-hard problem are not fairly satisfactory for online implementation in terms of performance-complexity trade-offs. This paper proposes a proximal gradient (PG) algorithm based on a kind of differential surrogate for the original nonsmooth objective. For better convergence rate, a fast proximal gradient (FPG) algorithm is also proposed, which guarantees convergence and offers state-of-the-art performance at low complexity. In each iteration, the FPG algorithm updates the solution along the gradient direction of the surrogate objective at a particular proximal point of the previous iteration and projects it back to the constraint set. Convergence of the PG/FPG iterates to a Karush-Kuhn-Tucker (KKT) point of the surrogate problem is established. Simulation results show that the FPG algorithm outperforms the existing SDP/SOCP-based methods and gradient-based iterative methods in terms of minimum signal-to-noise ratio (SNR) and computational time.