I Introduction
Various technological advances, including massive multipleinput multipleoutput (MIMO), millimeter wave (mmWave) communication, and ultradense network (UDN), have been proposed to enhance the spectrumefficiency of the fifthgeneration (5G) wireless networks and meet the everincreasing traffic demand [2]. However, these prominent technologies demand a large amount of radio frequency (RF) chains to operate at mmWave frequencies, which incur excessive energy consumption and lead to high implementation complexity in largescale cellular networks [3]. To realize a green and sustainable network evolution, it is imperative to develop new techniques that are not only spectrumefficient but also energyefficient [4].
Reconfigurable intelligent surface (RIS), as an emerging costeffective technology, has recently been proposed to enhance the spectrumefficiency and energyefficiency of wireless networks by reconfiguring the propagation environment [5, 6, 7]. A RIS is a metasurface composed of a large number of passive reflecting elements, each of which is able to independently change the phase shift of the incident signal to be reflected [8]. By adaptively altering the propagation of the reflected signal, the RIS is capable of achieving desired channel responses via constructive signal combination and destructive interference mitigation at the receivers, thereby enhancing the network performance [9]. With recent advancement on micro electromechanical systems and metamaterial [10], the phase shifts of the passive reflecting elements can be adjusted in a realtime manner, which makes RIS possible.
The research on RISempowered wireless networks has recently received considerable attention [11, 12, 13, 14, 15, 16, 17, 18, 19]. Specifically, the authors in [11] developed a resource allocation framework for RISempowered wireless networks, where the active beamforming at the base station (BS) and the passive beamforming at the RIS are jointly optimized to minimize the total transmit power. This work was then extended in [12] to consider the scenario that the RIS has discrete phase shifts. By taking into account both the transmit power and the hardware static power, the authors in [13] developed two efficient algorithms based on gradient descent search and fractional programming to maximize the energy efficiency of RISempowered wireless networks. Results in [11, 12, 13] showed that deploying a RIS in wireless networks has the potential to significantly reduce the power consumption. In addition, the authors in [14] developed three lowcomplexity algorithms to maximize the weighted sum rate of RISempowered multiuser networks. By further taking into account the user fairness, the authors in [15] proposed to maximize the minimum signaltointerferenceplusnoise ratio (SINR) of the users in the downlink. Moreover, by leveraging the elementwise optimization and alternating optimization, the authors in [16] studied the secrecy rate maximization problem for a simple scenario with one legitimate receiver and one eavesdropper. For the system with multiple legitimate receivers and multiple eavesdroppers, an iterative algorithm based on the alternating optimization and the pathfollowing algorithm was developed to solve the secrecy rate maximization problem in [17]. Besides, the RIS was also leveraged to boost the performance of overtheair computation [18] and energy harvesting [19].
Powerdomain nonorthogonal multiple access (NOMA), as another promising technique, can enhance the spectrumefficiency and improve the user fairness of 5G wireless networks by allowing the BS to serve multiple users in the same physical resource block (PRB) [20, 21]. The main idea of downlink powerdomain NOMA is to apply superposition coding at the BS and to perform successive interference cancellation (SIC) at the users except the one that is allocated the highest transmit power [22, 23]. The authors in [24] evaluated the performance of NOMA in practical cellular networks and showed that NOMA with appropriate user pairing and power allocation can achieve better performance than orthogonal multiple access (OMA). The performance gain of NOMA over OMA was also demonstrated in cooperative networks [25] and mmWave networks [26]. The achievable data rates of NOMA with channel condition and qualityofservice (QoS) based user ordering schemes were studied in [27] and [28], respectively, where a higher transmit power is allocated to the user with a worse channel condition and/or a higher QoS requirement. In addition, the authors in [29] and [30] studied the energyefficient beamforming design for NOMA networks and demonstrated the effectiveness of NOMA transmission in reducing the energy consumption. Until very recently, the application of RIS in NOMA networks was investigated in [31, 32, 33]. The authors in [31] analyzed the transmission reliability of RISaided NOMA transmission. For a multiuser singleinput singleoutput (SISO) NOMA network, the authors in [32] optimized the transmit power at the BS and the phase shifts at the RIS to enhance the user fairness. In addition, the authors [33] considered a RISaided downlink NOMA network, where both zeroforcing (ZF) beamforming and fixed user pairing were adopted to reduce the computational complexity at the cost of certain performance degradation.
In this paper, we consider the RISempowered multiuser multipleinput singleoutput (MISO) NOMA transmission of a downlink cellular network, where one BS serves multiple users with the assistance of the RIS. Both the transmit beamforming vectors at the BS and the phaseshift matrix at the RIS need to be jointly optimized to minimize the total transmit power, subjecting to the target data rate requirements and the unit modulus constraints. Such a joint optimization problem is highly intractable due to the nonconvex biquadratic constraints. We present an alternating optimization framework to decouple the optimization variables and reformulate the nonconvex quadratically constrained quadratic programming (QCQP) problems in each iteration as rankconstrained matrix optimization problems by leveraging matrix lifting. Although semidefinite relaxation (SDR)[11, 32, 34]
could be applied to solve this kind of lowrank optimization problems, the returned solution fails to satisfy the rankone constraints with a high probability, especially when the dimension of optimization parameters is high
[35, 18]. To address this issue, we propose an alternating differenceofconvex (DC) method, which can accurately detect the feasibility of the nonconvex rankone constraints. Furthermore, user ordering plays a critical role in determining the performance of NOMA. The design of user ordering is challenging in RISempowered multiuser NOMA networks, as the channel condition between the BS and each user depends not only on the direct and reflected channel responses but also on the phase shifts at the RIS and the target data rates of all users. Hence, we are motivated to develop a lowcomplexity user ordering scheme taking into account the concatenated channel responses, the phaseshifts, and the target data rates. The main contributions of this paper are summarized as follows:
We propose the joint design of the beamforming vectors and the phaseshift matrix to minimize the total transmit power, while taking into account the target data rate requirements of each user and the unit modulus constraints of each reflecting element. To reduce the implementation complexity, we also propose a lowcomplexity user ordering scheme, where the ordering criterion is derived in closedform and depends on both the concatenated channel conditions and the target data rates.

To address the limitations of the existing methods for nonconvex rankone constrained matrix optimization problems, we present an unified DC framework, where the rankone constraint is equivalently represented by the difference between the nuclear norm and the spectral norm. By inducing the rankone structure, the proposed alternating DC method is capable of accurately detecting the feasibility of nonconvex rankone constraints.

We develop an efficient alternating DC algorithm to iteratively update the beamforming vectors and the phase shifts in the lifted matrix space. By representing the objective functions of the resulting nonconvex DC programming problems in each iteration as the difference of two strongly convex functions, we prove that the proposed alternating DC algorithm always converges.

Through extensive simulations, we show that the total transmit power of the BS can be significantly reduced by deploying a RIS. The proposed alternating DC method considerably outperforms the existing methods, which demonstrates the superiority of the proposed algorithm. In addition, the proposed lowcomplexity user ordering scheme is numerically shown to have a comparable performance to the exhaustive search method. Results also show that the practical RIS structure with 3bit phase resolutions can achieve almost the same performance as the RIS with continuous phase shifts
The remainder of this paper is organized as follows. Section II describes the system model and problem formulation. We present an alternating optimization framework, and reformulate the nonconvex QCQP problem into the matrix optimization problem via matrix lifting in Section III. Section IV provides an alternating DC method, which iteratively solves the DC programming problems. In Section V, we present the overall algorithm and prove the convergence of the proposed algorithm. Section VI proposes a lowcomplexity user ordering scheme with closedform expressions. Section VII presents the numerical results. We conclude this paper in Section VIII.
Notations: Vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively. denotes the space of complexvalued matrices. denotes the statistical expectation. and denote the conjugate transpose and transpose, respectively. For a complexvalued vector , denotes its Euclidean norm and denotes a diagonal matrix with each diagonal entry being the corresponding element in . For a matrix , , , and denote its the Frobenius norm, the nuclear norm and the spectral norm, respectively. denotes the imaginary unit. denotes the real part of a complex number.
Ii System Model and Problem Formulation
Iia System Model
As shown in Fig. 1, we consider the RISempowered NOMA transmission of a downlink singlecell network, where a RIS with passive reflecting elements is deployed to assist the data transmission from an antenna BS to singleantenna users. To account for the increasing number of users, we consider an overloaded scenario, where the number of users is more than the number of antennas at the BS, i.e.,
. The RIS can be switched between two operational modes, i.e., a receiving mode for channel state information (CSI) estimation and a reflecting mode for incident signal scattering
[36]. We denote and as the signal and linear beamforming vector at the BS for user , respectively, where with . Without loss of generality, signalis assumed to have zero mean and unit variance, i.e.,
. In addition, the quasistatic flatfading model is adopted for all channels. To characterize the theoretical performance gain brought by the RIS, we assume that the CSI of all channels are perfectly known at the BS [11, 12, 13, 14, 15, 16, 17, 18, 19]. With universal frequency reuse, the signal received at user , after transmitted by the BS and reflected by the RIS, is given by(1) 
where , and denote the channel responses from the BS to user , from the BS to the RIS, and from the RIS to user , respectively, and is the additive white Gaussian noise (AWGN) with being the noise power. In addition, denotes the diagonal phaseshift matrix of the RIS, where and denote the phase shift of element and the amplitude reflection coefficient on the incident signal, respectively. As each element on the RIS is designed to maximize the signal reflection, we assume without loss of generality[11, 12, 13, 15, 16, 17, 18, 19]. Due to the severe path loss, the power of signals that are reflected by the RIS two or more times is assumed to be negligible [11, 12, 13, 14, 15, 16, 17, 18, 19].
We consider the multiuser powerdomain NOMA transmission in the downlink. According to the decoding principle of NOMA, each user performs SIC to remove the signal(s) being allocated a higher transmit power than its own signal [37]. It has been shown in [27] that the decoding order plays an important role in determining the performance of NOMA and is affected by both the transmit power allocation and the channel conditions. However, in RISempowered NOMA networks, the concatenated channel response depends on , , , and , which significantly complicates the user ordering. As there are different decoding orders for users, we denote the set of all possible user orderings as , where denotes the th decoding order. For a given user ordering , index refers to the user which is allocated the th highest transmit power. Specifically, user directly decodes its own signal by treating the signals intended for other users as noise. On the other hand, user , , sequentially decodes and removes signals , until its own signal is decoded. After successfully removing the signals intended for users , the remaining signal at user , where , can be expressed as
(2) 
As signal is required to be decoded by users , the achievable data rate of user can be written as
(3) 
where the channel bandwidth is normalized to 1 and the SINR of signal observed at user can be expressed as
(4) 
IiB Problem Formulation
In this subsection, we formulate a total transmit power minimization problem by jointly optimizing the beamforming vectors (i.e., ) at the BS and the phaseshift matrix (i.e., ) at the RIS, taking into account the data rate requirements of all users and the unit modulus constraints of all reflecting elements. For a given decoding order , we denote the optimal total transmit power as , where denotes the solution of the following optimization problem. For notational ease, we omit the decoding order index in the sequel. The transmit power minimization problem is formulated as
subject to  (6)  
where is the transmit power allocated to user and denotes the minimum data rate requirement of user . To assist the algorithm design, we rewrite constraints of problem as
(7) 
where is the minimum SINR required to successfully decode signal . Constraints can be further rewritten as
(8) 
Therefore, problem can be equivalently rewritten as
subject to  (9) 
However, problem is still highly intractable due to the nonconvex biquadratic constraints (8), in which the beamforming vectors and the phaseshift matrix are coupled. To address this challenge, we present an alternating optimization framework to solve problem in Sections III, IV, and V. The optimal total transmit power can be obtained by exhaustively searching over all possible decoding orders, i.e., . To reduce the computational complexity, we develop a lowcomplexity user ordering scheme in Section VI, which is shown to achieve almost the same performance as the exhaustive search method in Section VII.
Iii Alternating Optimization Framework
In this section, we present an alternating optimization framework to solve problem . In particular, the beamforming vectors and the phaseshift matrix are optimized alternatively until convergence. Moreover, we transform the resulting nonconvex QCQP problem in each iteration of the alternating optimization into a rankconstrained matrix optimization problem via matrix lifting.
Iiia Beamforming Vectors Optimization
For a given phaseshift matrix , the concatenated channel response is fixed, and hence problem can be simplified as the following nonconvex QCQP problem
subject to  (10) 
Problem is a nonconvex problem due to the nonconvexity of the quadratic constraints. To address these nonconvex constraints, a natural way is to reformulate problem as a semidefinite programming (SDP) problem by using the SDR technique [11, 34]. This is achieved by transforming the resulting nonconvex QCQP problem into a rankconstrained matrix optimization problem via matrix lifting, following by dropping the rankone constraints. If the returned solution of the resulting SDP problem does not satisfy the rankone constraints, then Gaussian randomization technique [34] is adopted to obtain a suboptimal solution. Specifically, by lifting vector into a positive semidefinite (PSD) matrix with , , problem can thus be equivalently rewritten as
subject to  (11)  
where .
IiiB PhaseShift Matrix Optimization
When beamforming vectors are given, we denote and , . Hence, we have , where . Thus, problem can be simplified as the following nonconvex feasibility detection problem
subject to  (12)  
Problem is nonconvex and inhomogeneous due to the nonconvexity of the quadratic constraints and the unit modulus constraints. Fortunately, it can be reformulated as a homogenous nonconvex QCQP problem by introducing an auxiliary variable . We rewrite problem as
subject to  (13)  
where
(14) 
If we obtain a feasible solution, denoted as , to problem , then a feasible solution to problem can immediately be obtained by setting , where denotes the first elements of vector . Similarly, the matrix lifting technique is applied to reformulate the nonconvex quadratic constraints in problem . By denoting and , problem can be equivalently rewritten as the following rankone constrained matrix optimization problem
subject to  (15)  
Although the SDR technique can solve problems and , the probability of the obtained solution satisfying the rankone constraints becomes small in the high dimensional setting, yielding performance degradation [35, 18]. To address the limitations of the SDR technique, we shall propose an exact DC representation for the rankone constraint of the PSD matrix by exploiting the difference between the nuclear norm and the spectral norm in the following section.
Iv The Framework of Alternating DC Method
In this section, we present an exact DC representation for the rankone constraints, followed by proposing an alternating DC method to solve the original rankconstrained matrix optimization problems.
Iva DC Representation for RankOne Constraint
For a matrix , the rankone constraint can be equivalently rewritten as
(16) 
where is the
th largest singular value of matrix
, and is the norm of a vector, i.e., the number of nonzero entries. Note that the nuclear norm and the spectral norm are, respectively, given by(17) 
We introduce an exact DC representation for the rankone constraint in the following proposition.
Proposition 1.
For a PSD matrix with , we have
. If matrix is a rankone PSD matrix, then the nuclear norm is equal to the spectral norm as for all . Hence, implies that
(18) 
Because of , we have . Therefore, is equivalent to .
It is noteworthy that the rank (i.e., ()) is a discontinuous function, whereas the DC representation (i.e., ) is a continuous function. Moreover, both and are convex functions [38].
IvB Proposed Alternating DC Method
The main idea of our proposed alternating DC method is to first apply the DC representation to reformulate problems and , and then alternatively solve two DC programming problems until convergence.
Given the phaseshift matrix , we solve the following DC programming problem to find rankone matrices for problem
subject to  (19)  
where is a penalty parameter. By enforcing the penalty term to be zero, problem induces rankone matrices. After solving problem (IVB), we can recover the beamforming vectors through Cholesky decomposition, i.e., , , where denotes the solution of problem .
Similarly, given beamforming vectors , we minimize the following difference between the nuclear norm and the spectral norm to detect the feasibility of problem
subject to  (20)  
Specifically, when the objective value of problem becomes zero, we obtain an exact rankone feasible solution, denoted as . Using Cholesky decomposition , we obtain a feasible solution to problem and in turn obtain a feasible solution to problem (IIIB) as .
V Alternating DC Algorithm with Convergence Guarantee
In this section, we propose an efficient alternating DC algorithm to obtain the high quality solutions for beamforming vectors and the phaseshift matrix with convergence guarantee.
Va An Unified Difference of Strongly Convex Functions Representation
Although the DC programming problems and are still nonconvex, they have a good structure that can be exploited to develop an efficient algorithm, which successively solves the convex relaxation of the primal and dual problems of DC programming [39]. In order to establish some important properties of the algorithm, we represent the DC objective function as the difference of two strongly convex functions. Specifically, we equivalently rewrite problem as
(21) 
and problem as
(22) 
where and denote the PSD cones that satisfy the constraints in problems and , respectively, and the indicator function is defined as
We rewrite the DC functions and as the difference of two strongly convex functions, i.e., and , where
(23)  
(24)  
(25)  
(26) 
Because of the additional quadratic terms (i.e., and ), , , , and are all strongly convex functions. It turns out that problems and have the unified structure of minimizing the difference of two strongly convex functions, i.e.,
(27) 
To solve the nonconvex DC programming problem, we present a dualitybased DC algorithm to construct a sequence of candidates to the primal and dual solutions in the following subsection.
VB DualityBased DC Algorithm for Problem
According to the Fenchel’s duality [40], the dual problem of problem is represented by
(28) 
where and are the conjugate functions of and , respectively. The conjugate function is defined as
(29) 
where the inner product is defined as according to Wirtinger’s calculus [41] in the complex domain and denotes ’s feasible solution region. Since the primal problem and its dual problem are still nonconvex, the dualitybased DC algorithm iteratively updates both the primal and dual variables via successive convex approximation. In the th iteration, we have
(30)  
(31) 
Based on the Fenchel biconjugation theorem [40], can be represented as
(32) 
where is the subgradient of with respect to at . Thus, at the th iteration for problem can be obtained by solving the following convex optimization problem
subject to  (33)  
Similarly, at the th iteration for problem can be obtained by solving the following convex optimization problem
subject to  (34)  
Problems and are convex and can be efficiently solved by using CVX [42]. The and are, respectively, given by
(35)  
(36) 
It is worth noting that the subgradient of at (i.e., ) can be efficiently computed according to the following proposition.
Proposition 2.
For a PSD matrix , the subgradient of at can be efficiently computed as , where
is the eigenvector corresponding to the largest eigenvalue
.The computationally efficient dualitybased DC algorithm with convergence guarantee is developed by successively solving the convex relaxation of the primal and dual problems of DC programming. The overall algorithm, solving problems and in an alternative manner, is referred as the alternating DC algorithm, which is summarized in Algorithm 1. Specifically, Algorithm 1 optimizes and alternatively, where the presented dualitybased DC algorithm is adopted to obtain the beamforming vectors and the phase shifts in the lifted matrix space that satisfy the rankone constraints. The alternating DC algorithm terminates when the decrease of the objective value of problem is smaller than , which is a predetermined convergence threshold. Moreover, we prove the convergence of Algorithm 1 in the following subsection.
VC Convergence Analysis of Proposed Alternating DC Algorithm
Before proving the convergence of the proposed alternating DC algorithm, we present some important properties of the solutions obtained by solving the convex relaxation of the primal and dual problems of DC programming in the following proposition.
Proposition 3.
For any the sequence generated by iteratively solving problem
has the following properties:
(i) The sequence converges to a stationary point of in from an arbitrary initial point, and the sequence is strictly decreasing and convergent.
(ii) For any we have
(37) 
where is the global minimum of and denotes the average of the sequence .
Likewise, for any , the sequence generated by iteratively solving problem
has the following properties:
(iii) The sequence converges to a stationary point of in from an arbitrary initial point, and the sequence of is strictly decreasing and convergent.
(iv) For any we have
(38) 
where is the global minimum of .
Please refer to Appendix A.
Based on Proposition 3, the convergence analysis of Algorithm 1 is presented in the following proposition.
Proposition 4.
The objective value of problem in is decreasing over iterations until convergence by applying the proposed alternating DC algorithm.
We denote as the objective value of problem for a feasible solution . We denote as a feasible solution of problem at the th iteration. For a given , we apply the presented dualitybased DC algorithm to obtain a solution for problem , based on which we obtain as the intial point for the iteration. Because the presented dualitybased DC algorithm can accurately detect the feasibility of rankone constraints, the solution can be obtained via cholesky decomposition, where . Hence, we have and . According to Proposition 3, the object value of problem is strictly decreasing over iterations. Hence, we have
(39) 
Based on Algorithm 1, we have
(40) 
For a given , we also apply the dualitybased DC algorithm to solve problem . Based on Algorithm 1, if there exists a feasible solution to problem , it is also feasible to problem , i.e., exists. It follows that
(41) 
where the equality holds as the value of is independent of but only depends on . Based on and , we further have
(42) 
According to , the objective value of problem is always decreasing over iterations. Therefore, the proposed alternating DC algorithm converges, which completes the proof.
Vi LowComplexity User Ordering Scheme
Solving optimization problem times by exhaustive search to obtain the optimal total transmit power is computational prohibitive. In this section, we develop a lowcomplexity user ordering scheme to determine the decoding order of the users.
In general, the BS allocates a higher transmit power to the user that has a lower channel gain toward the BS and/or a higher target data rate, which can achieve an excellent performance[43]. Motivated by this observation, we order the users according to the minimum transmit power required to meet the target data rate requirement of each user, when the intracell interference is temporarily ignored. Specifically, the minimum transmit power required at the BS to serve user can be obtained by solving the following problem
subject to  (43)  
Note that problem is a nonconvex optimization problem due to the coupled the optimization variables and . To tackle the nonconvexity, we leverage the alternating optimization technique presented in Section III to decouple and .
For a given phaseshift matrix , it can be verified that the optimal transmit beamforming solution to problem can be obtained by maximumratio transmission (MRT) [44]. As a result, we have
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