# Social-Aware-Based Resource Allocation for NOMA-Enhanced D2D Communications
## 3 Social-Aware-Based Resource Allocation
### Optimization Model
#### 3.2.2 Power Control  
In this section, we need to solve the power allocation problem for D2D content providers according to the channel allocation result \(\eta^{*}\). The power allocation problem can be described as  
\[P_{2} : \max_{P_{d}^{k}}\ \sum_{k\in[D]}R_{d}^{k}\Big{(}\eta^{*},P_{d}^{k} \Big{)}\] (30) s.t. \[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\] \[\gamma_{d}^{k}\geq\gamma_{d}^{th}\quad k\in\{D\}\] (32) \[0\leq P_{d}^{k}\leq P_{\max}\quad\forall k \tag{33}\]  
The optimization problem is a nonconvex NP-hard problem because of the interference between D2D clusters. So as to reduce the computational complexity, we use SCA [13] to transform the optimization problem into a convex problem, and find the suboptimal solution of it. Equation (30) can be rewritten as  
\[\max_{P_{d}^{k}}\sum_{k\in[D]}\log_{2}\Big{(}1+\gamma_{d}^{k}\Big{)} \tag{34}\]  
The optimization problem conforms to the following inequality according to [14]:  
\[\log_{2}\Big{(}1+\gamma_{d}^{k}\Big{)}\geq\alpha_{d}^{k}\log\gamma_{d}^{k}+ \beta_{d}^{k} \tag{35}\]  
where \(\alpha_{d}^{k}\) and \(\beta_{d}^{k}\) are defined as  
\[\alpha_{d}^{k}=\frac{\gamma_{d}^{k}}{1+\gamma_{d}^{k}} \tag{36}\]  
\[\beta_{d}^{k}=\log\Big{(}1+\gamma_{d}^{k}\Big{)}-\frac{\gamma_{d}^{k}}{1+ \gamma_{d}^{k}}\log\gamma_{d}^{k} \tag{37}\]  
The equality is satisfied when \(\gamma_{d}^{k}=\gamma_{d}^{k}\). Then the lower bound of the optimization problem can be represented as  
\[\sum_{k\in[D]}\log_{2}\Big{(}1+\gamma_{d}^{k}\Big{)}\geq f\Big{(}P_{d}^{k} \Big{)} \tag{38}\]  
where \(f\Big{(}P_{d}^{k}\Big{)}\) can be defined as  
\[f\Big{(}P_{d}^{k}\Big{)}=\sum_{k\in[D]}\Big{(}\alpha_{d}^{k}\log_{2}\gamma_{d }^{k}+\beta_{d}^{k}\Big{)} \tag{39}\]The original optimization problem can be transformed into maximizing its lower bound. We set \(P_{d}^{k}=2^{\beta_{d}^{k}}\) in order to transform the optimization problem into a convex problem. The original optimization problem can be rewritten as  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]0  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]1  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]2  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]3  
In the transformed optimization problem, both the optimization goal and the constraint conditions are formed by log-sum-exp function. As the log-sum-exp function is a convex structure [15], the transformed optimization problem is a convex problem. In order to prove the convexity of the optimization problem (Equation (40)), we can make the following proposition. **Proposition 1**. : _Assume that there are two D2D clusters in the cell reusing the same cellular channel. One D2D cluster contains one D2D content requester and one D2D content provider. They communicate via one-to-one D2D communication. The other contains one D2D content requester and two D2D content providers. They communicate via many-to-one D2D communication with NOMA in \[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]4 In such scenario, the optimization problem (Equation (40)) is convex._  
**Proof. ** The data rate of D2D content provider \(DT_{1}\), \(DT_{2}\), and \(DT_{3}\) can be represented as  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]5  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]6  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]7  
\[\gamma_{\mathrm{m}}\geq\gamma_{m}^{th}\quad\forall m \tag{31}\]8 System model of Proposition 1.