  This paper studies the transmit power optimization in multi-cell massive
multiple-input multiple-output (MIMO) systems. Network-wide max-min fairness
(NW-MMF) and network-wide proportional fairness (NW-PF) are two well-known
power control schemes in the literature. The NW-MMF focus on maximizing the
fairness among users at the cost of penalizing users with good channel
conditions. On the other hand, the NW-PF focuses on maximizing the sum SE,
thereby ignoring fairness, but gives some extra attention to the weakest users.
However, both of these schemes suffer from a scalability issue which means that
for large networks, it is highly probable that one user has a very poor channel
condition, pushing the spectral efficiency (SE) of all users towards zero. To
overcome the scalability issue of NW-MMF and NW-PF, we propose a novel power
control scheme that is provably scalable. This scheme maximizes the geometric
mean (GM) of the per-cell max-min SE. To solve this new optimization problem,
we prove that it can be rewritten in a convex optimization form and then solved
using standard tools. The simulation results highlight the benefits of our
model which is balancing between NW-PF and NW-MMF.
