Reconfigurable intelligent surface (RIS) has recently been recognized as a promising technology that can enhance the energy-efficiency and spectrum-efficiency of wireless networks. In this paper, we consider an RIS-empowered downlink non-orthogonal multiple access (NOMA) network, where the beamforming vectors at the BS and the phase-shift matrix at the RIS are jointly optimized to minimize the total transmit power, taking into account the user ordering, the users' data rate requirements, and the reflecting elements' unit modulus constraints. However, the formulated problem is highly intractable due to non-convex bi-quadratic constraints. To this end, we present an alternating optimization framework to decouple the optimization variables and transform the quadratic programming problem in each alternating procedure into a fixed-rank matrix optimization problem via matrix lifting. By exploiting the difference between the nuclear norm and the spectral norm, we then propose a difference-ofconvex (DC) function representation for the rank function to accurately detect the feasibility of the non-convex rank-one constraints. Moreover, we develop a novel alternating DC algorithm to solve the resulting DC programming problems and prove the convergence of the proposed algorithm. To reduce the implementation complexity, we further propose a low-complexity user ordering scheme, where the ordering criterion is derived in closed-form. Simulation results demonstrate the effectiveness of deploying an RIS and the superiority of the proposed alternating DC algorithm in reducing the total transmit power. Index TermsReconfigurable intelligent surface, non-orthogonal multiple access, joint beamformer and phaseshift matrix optimization, non-convex bi-quadratic optimization, difference-of-convex programming.

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