  In this paper, we study the physical layer multicasting to multiple
co-channel groups in large-scale antenna systems. The users within each group
are interested in a common message and different groups have distinct messages.
In particular, we aim at designing the precoding vectors solving the so-called
quality of service (QoS) and weighted max-min fairness (MMF) problems, assuming
that the channel state information is available at the base station (BS). To
solve both problems, the baseline approach exploits the semidefinite relaxation
(SDR) technique. Considering a BS with $N$ antennas, the SDR complexity is more
than $\mathcal{O}(N^{6})$, which prevents its application in large-scale
antenna systems. To overcome this issue, we present two new classes of
algorithms that, not only have significantly lower computational complexity
than existing solutions, but also largely outperform the SDR based methods.
Moreover, we present a novel duality between transformed versions of the QoS
and the weighted MMF problems. The duality explicitly determines the solution
to the weighted MMF problem given the solution to the QoS problem, and vice
versa. Numerical results are used to validate the effectiveness of the proposed
solutions and to make comparisons with existing alternatives under different
operating conditions.
