## 2 System Model and Problem Formulation
### Problem Formulation  
In this paper, our goal is to gain minimal D2D power consumption with maximal system capacity while satisfying the QoS requirement of CUs and D2D users. Hence, we introduce weighted coefficients for power and capacity to strike and regulate the balance between D2D links' aggregated power consumption minimization and system capacity maximization. For simplicity, we denote the SINR thresholds of cellular links and D2D links as \(\Gamma_{k}^{C}\) and \(\Gamma_{m}^{D}\), which must be attained to meet the individual QoS requirement. In addition, we use \(\mathcal{S}_{m}=\left\{k\in\mathcal{K}\middle|\omega_{m,k}=1\right\}\) to denote the set of the sub-channels which are assigned to the \(m\)-th multicast group. Then, by using the linear weighted aggregative method, we formulate the multi-objective optimization problem as  
\[\min_{\{p_{k}^{C},p_{m,k}^{D},\omega_{m,k}\}}\sum_{m\in\mathcal{M}}\sum_{k\in \mathcal{K}}\omega_{m,k}\left\{\frac{\eta\cdot p_{m,k}^{D}}{p_{m,k,\max}^{D}}- \frac{(1-\eta)\cdot X_{n,m,k}}{|\mathcal{D}_{m}|}\right\}, \tag{5}\]  
\[\left\{\begin{array}{l}p_{m,k}^{D}(\gamma_{k,m}^{C\,*},\gamma_{n,m,k}^{D})=\frac{ \left(H_{k,B}^{CB}+\widetilde{H}_{k,m,n}^{CD}\Gamma_{k}^{C}\right)\gamma_{n,m,k} ^{D}\sigma_{0}^{2}}{H_{k,B}^{CB}H_{m,n}^{DD}-\widetilde{H}_{m,B}^{DB}\widetilde{ H}_{k,m,n}^{CD}\Gamma_{k}^{C}\gamma_{n,m,k}^{D}}\\ p_{k}^{C}(\gamma_{k,m}^{C\,*},\gamma_{n,m,k}^{D})=\frac{\left(H_{m,n}^{C}+ \widetilde{H}_{k,m,n}^{CD}\gamma_{n,m,k}^{D}\right)\Gamma_{k}^{C}\sigma_{0}^{ 2}}{H_{k,B}^{CB}H_{m,n}^{DD}-\widetilde{H}_{m,B}^{DB}\widetilde{H}_{k,m,n}^{CD }\Gamma_{k}^{C}\gamma_{n,m,k}^{D}}\end{array}\right.. \tag{9}\]  
Then, we substitute \(p_{m,k}^{D}(\gamma_{k,m}^{C\,*},\gamma_{n,m,k}^{D})\) into Equation (5), and the optimization problem can be rewritten as  
\[\min_{\{\gamma_{n,m,k}^{D},\omega_{m,k}\}}\sum_{m\in\mathcal{M}}\sum_{k\in \mathcal{K}}\omega_{m,k}\left\{\eta\frac{p_{m,k}^{D}(\gamma_{k,m}^{C\,*}, \gamma_{n,m,k}^{D})}{P_{m,k,\max}^{D}}-(1-\eta)\frac{X_{n,m,k}}{|\mathcal{D}_{ m}|}\right\}. \tag{10}\]  
Define  
\[\Phi\left(\gamma_{k,m}^{C\,*},\gamma_{n,m,k}^{D}\right)=\eta\frac{p_{m,k}^{D} (\gamma_{k,m}^{C\,*},\gamma_{n,m,k}^{D})}{P_{m,k,\max}^{D}}-(1-\eta)\frac{X_{ n,m,k}}{|\mathcal{D}_{m}|}. \tag{11}\]  
Then \(\mathrm{Ea}\)ution (10) can be expressed as  
\[\min_{\{\gamma_{n,m,k}^{D},\omega_{m,k}\}}\sum_{m\in\mathcal{M}}\sum_{k\in \mathcal{K}}\omega_{m,k}\Phi\left(\gamma_{k,m}^{C\,*},\gamma_{n,m,k}^{D}\right). \tag{12}\]  
As explained above, if the resource of CU \(k\) is reused by D2D group \(m\), the constraints C5, C6 and C7 should be satisfied. Then, we introduce the following Proposition.  
**Proposition 1**.: _The optimal solutions of the power values \(p_{m,k}^{D}\) of the optimization problem in Equation (12) must be the optimal solutions of the optimization problem \(\min\limits_{\{\gamma_{n,m,k}^{D}\}}\sum\limits_{k\in\mathcal{S}_{m}}\Phi \left(\gamma_{k,m}^{C\,*},\gamma_{n,m,k}^{D}\right)\)._  
Proof.: The proof is presented in Appendix A.  
It can be seen from Proposition 1 that the original dimensional non-convex optimization problem can be transformed into two separate optimization sub-problems. In the first sub-problem, for single D2D group with a given multiple sub-channels reusing, we find the optimal SINR value on each sub-channel to achieve the multi-objective optimization. In the second sub-problem, for multiple D2D groups sharing the resources with multiple CUs, we propose a practice algorithm to allocate each sub-channel to the D2D multicast groups.