  Consider a multiuser downlink beamforming optimization problem for the
non-orthogonal multiple access (NOMA) transmission in a multiple-input
single-output (MISO) system. The total transmission power minimization problem
is formulated subject to both per-antenna power constraints and
quality-of-service (QoS) constraints under the NOMA principal. The problem is a
non-convex quadratically constrained quadratic program, and the conventional
semidefinite program (SDP) relaxation is not tight. In order to tackle the
non-convex NOMA beamforming problem, we construct a second-order cone
approximation for each signal-to-interference-plus-noise ratio (SINR)
constraint and form an iterative algorithm, in which a sequence of second-order
cone programs (SOCPs) are solved. The optimal values of the sequence of SOCPs
are non-increasing, and it converges to a locally optimal value. However, our
extensive simulation results show that the locally optimal value is more or
less as good as the globally optimal value. In particular, we show that the SDP
relaxation is tight for two-user case if one of the SINR constraints is strict
(non-binding) at the optimality. Detailed simulation results are presented to
demonstrate the performance gains of the NOMA downlink beamforming with
per-antenna power constraints through the proposed approximate algorithm.
