  This paper considers downlink of reconfigurable intelligent surface (RIS)
assisted cooperative non-orthogonal multiple access (CNOMA) systems. Our
objective is to minimize the total transmit power by jointly optimizing the
active beamforming vectors, transmit-relaying power, and RIS phase shifts. The
formulated problem is a mixed-integer nonlinear programming (MINLP) problem. To
tackle this problem, the alternating optimization approach is utilized to
decouple the variables. In each alternative procedure, the optimal solutions
for the active beamforming vectors, transmit-relaying power and phase shifts
are obtained. However, the proposed algorithm has high complexity since the
optimal phase shifts are solved by integer linear programming (ILP) whose
computational complexity is exponential in the number of variables. To strike a
good computational complexity-optimality trade-off, a low-complexity suboptimal
algorithm is proposed by invoking the iterative penalty function based
semidefinite programming (SDP) and the successive refinement approaches.
Numerical results illustrate that: i) the proposed RIS-CNOMA system, aided by
our proposed algorithms, outperforms the conventional CNOMA system. ii) the
proposed low-complexity suboptimal algorithm can achieve the near-optimal
performance. iii) whether the RIS-CNOMA system outperforms the RIS assisted
non-orthogonal multiple access (RIS-NOMA) system depends not only on the users'
locations but also on the RIS' location.
