## 2 System Model and Problem Formulation  
Consider a downlink NOMA system that serves \(n\) users, as shown in Figure 1. The maximum transmission power of the base station (BS) for the downlink NOMA system is recorded as \(P_{B}\). The power allocated to the \(i\)-th user is denoted as \(P_{i}\), where \(i=1,2,\cdots,n\) and  
\[P_{1}+P_{2}+\cdots+P_{n}\leq P_{B}. \tag{1}\]  
The parameter \(H_{i}\) represents the normalized channel gain between the BS and the \(i\)-th user, where \(i=1,2,\cdots,n\), and  
\[H_{1}\geq H_{2}\geq\cdots\geq H_{n}. \tag{2}\]  
Using SIC in the NOMA systems, multiple users with different channel gain can be multiplexed together into clusters and transmit in the same subchannel. The basic principle of SIC is to gradually subtract the users' interference with the maximum signal power gain. Concretely, the BS broadcasts the superposition of signals to all the users in its servicing cluster via power-domain division, where users are sorted on each subchannel in descending order of power gains [47]. After receiving the transmission signal from the subchannel, SIC is used to eliminate multi-user interference. In the received signal, the SIC detector decides the data of multiple users one by one and subtracts the multiple access interference caused by the user's signal at the same time. In practical applications, to ensure the NOMA systems' overall performance, users with lower channel gain will have higher power gain. Therefore, the SIC detector generally operates in the order of channel gain from low to large, where the signal with the smallest channel gain will be decided first. Formally, the \(j\)-th user first decodes the messages of \(j-1\) users after it, i.e., the message of the \(i\)-th user where \(i>j\), and then removes these messages from the received superposed signal in the order of \(i=n,n-1,\cdots,j\). Conversely, other remaining messages, i.e., the message of the \(i\)-th user where \(i<j\), are treated as noise [48; 49]. In this way, the loop operation is carried out until all of the multiple access interference is eliminated. By these steps, all users can get their desired signal. In such a scenario, the total system bandwidth \(B\) and the maximum transmission power \(P_{B}\) are evenly divided into \(m\) frequency resource blocks with the bandwidth \(b=\frac{B}{m}\) and the power budge \(p_{b}=\frac{P_{B}}{m}\) for each resource block. Here, the number of resource blocks \(m\) is a preset parameter in the NOMA systems. After clustering \(n\) users into \(c\) clusters where \(1\leq c\leq\frac{n}{2}\), \(\omega\), resource blocks are allocated to the \(j\)-th cluster where \(0\leq\omega_{j}\leq m\) and \(\omega_{1}+\omega_{2}+\cdots+\omega_{c}=m\). The same cluster users can share resource blocks by non-orthogonal scheduling while different clusters can work independently. For a general \(n\)-user downlink NOMA system, the necessary power constraints for efficient SIC [50] can be expressed as  
\[H_{i-1}P_{i}-\sum_{j=1}^{i-1}H_{i-1}P_{j}\geq P_{T} \tag{3}\]  
where \(i=2,3,\cdots,n\), and \(P_{T}\) are the minimum power difference requirements for signal decoding from the remaining non-decided message signals. According to the Shannon-Hartley theorem [51], the achievable throughput for the \(i\)-th of all users assigned to the \(j\)-th cluster can be expressed as  
\[\omega_{j}b\log_{2}\left(1+\frac{H_{i}P_{i}}{\omega_{j}+\sum_{k=1}^{i-1}H_{i}P _{k}}\right). \tag{4}\]  
Let an \(m\times c\) matrix \(\delta\) whose entry in the \(i\)-th row and the \(j\)-th column be given as \(\delta_{ij}\), denoting a result of clustering, i.e., \(\delta_{ij}=1\) means the \(i\)-th user is grouped into the \(j\)-th cluster while \(\delta_{ij}=0\) is not. Let  
\[P=\{P_{1},P_{2},\cdots,P_{n}\} \tag{5}\]  
Figure 1: A downlink NOMA system model. denote the power allocated to each user. Denote the minimum data transmission rate required by the quality of service (QoS) of the \(i\)-th user as \(R_{i}\).