  We propose a first-order fast algorithm for the weighted max-min fair (MMF)
multi-group multicast beamforming problem in large-scale systems. Utilizing the
optimal multicast beamforming structure obtained recently, we convert the
nonconvex MMF problem into a min-max weight minimization problem and show that
it is a weakly convex problem. We propose using the projected subgradient
algorithm (PSA) to solve the problem directly, instead of the conventional
method that requires iteratively solving its inverse problem. We show that PSA
for our problem has closed-form updates and thus is computationally cheap.
Furthermore, PSA converges to a near-stationary point of our problem within
finite time. Simulation results show that our PSA-based algorithm offers
near-optimal performance with considerably lower computational complexity than
existing methods for large-scale systems.
