  In this paper, we consider a two-hop one-way relay network for multigroup
multicast transmission between long-distance users, in which the relay is
equipped with multiple antennas, while the transmitters and receivers are all
with a single antenna. Assuming that perfect channel state information is
available, we study amplify-and-forward (AF) schemes that aim at optimizing the
max-min-fair (MMF) rate. We begin by considering the classic beamformed AF
(BF-AF) scheme, whose corresponding MMF design problem can be formulated as a
rank-constrained fractional semidefinite program (SDP). We show that the gap
between the BF-AF rate and the SDR rate associated with an optimal SDP solution
is sensitive to the number of users as well as the number of power constraints
in the relay system. This reveals that the BF-AF scheme may not be well suited
for large-scale systems. We therefore propose the stochastic beamformed AF
(SBF-AF) schemes, which differ from the BF-AF scheme in that time-varying AF
weights are used. We prove that the MMF rates of the proposed SBF-AF schemes
are at most $0.8317$ bits/s/Hz less than the SDR rate, irrespective of the
number of users or power constraints. Thus, SBF-AF can outperform BF-AF
especially in large-scale systems. Finally, we present numerical results to
demonstrate the viability of our proposed schemes.
