  In this paper, a downlink multiple input multiple output (MIMO)
non-orthogonal multiple access (NOMA) wireless communication system is
considered. In NOMA systems, the base station has unicast data for all users,
and multiple users in a group share the same resources. The objective is to
design transmit precoders and power allocation coefficients jointly that
provide max-min fairness (MMF) among the strongest users in each group, while
maintaining minimum target rates for all the other users. The problem is solved
via two main iterative approaches. The first method is based on semi-definite
relaxation (SDR) and successive convex approximation (SCA), and the second
method is based on the equivalency between achievable rate and minimum mean
square error (MMSE) expressions. For the latter approach, Karush-Kuhn-Tucker
(KKT) optimality conditions are derived and the expressions satisfied by the
optimal receivers, MMSE weights and the optimal precoders are obtained.
Proposed algorithms are compared with rate-splitting (RS), orthogonal multiple
access (OMA) and multi-user linear precoding (MULP) schemes in terms of MMF
rates, energy efficiency and complexity. It is shown that while RS has the best
MMF rates and energy efficiency, the MMSE approach based on KKT optimality
conditions has the least complexity. Moreover, the SDR/SCA approach offers an
excellent tradeoff. It offers high MMF rates, low complexity and superior
energy efficiency.
