  This work investigates the performance of intelligent reflective surfaces
(IRSs) assisted uplink non-orthogonal multiple access (NOMA) in
energy-constrained networks. Specifically, we formulate and solve two
optimization problems; the first aims at minimizing the sum of users' transmit
power, while the second targets maximizing the system level energy efficiency
(EE). The two problems are solved by jointly optimizing the users' transmit
powers and the beamforming coefficients at IRS subject to the users' individual
uplink rate and transmit power constraints. A novel and low complexity
algorithm is developed to optimize the IRS beamforming coefficients by
optimizing the objective function over a \textit{complex circle manifold}
(CCM). To efficiently optimize the IRS phase shifts over the manifold, the
optimization problem is reformulated into a feasibility expansion problem which
is reduced to a max-min signal-plus-interference-ratio (SINR). Then, with the
aid of a smoothing technique, the exact penalty method is applied to transform
the problem from constrained to unconstrained. The proposed solution is
compared against three semi-definite programming (SDP)-based benchmarks which
are semi-definite relaxation (SDR), SDP-difference of convex (SDP-DC) and
sequential rank-one constraint relaxation (SROCR). The results show that the
manifold algorithm provides better performance than the SDP-based benchmarks,
and at a much lower computational complexity for both the transmit power
minimization and EE maximization problems. The results also reveal that
IRS-NOMA is only superior to orthogonal multiple access (OMA) when the users'
target achievable rate requirements are relatively high.
