## 2 System Model and Problem Formulation
### SINR Assignment for One D2D Multicast Group
Then, we write its Lagrangian function as  
\[L\left(\left\{\gamma^{D}_{n,m,k}\right\},\lambda\right)=\sum\limits_{k\in \mathcal{S}_{m}}\Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)+ \lambda\left[\log_{2}(1+\Gamma^{D}_{m})-\sum\limits_{k\in\mathcal{S}_{m}}\log _{2}(1+\gamma^{D}_{n,m,k})\right]. \tag{18}\]  
By applying Karush-Kuhn-Tucker (KKT) conditions [14], the first order optimality conditions and the equation constraint are given as follows,  
\[\frac{\partial L\left(\left\{\gamma^{D}_{n,m,k}\right\},\lambda \right)}{\partial\gamma^{D}_{n,m,k}} =\frac{\eta\sigma^{2}_{0}H^{CB}_{k,B}H^{DD}_{m,n}\left(H^{CB}_{k,B} +\widetilde{H}^{CD}_{k,m,n}\Gamma^{C}_{k}\right)}{P^{D}_{m,k,\max}\left(H^{CB }_{k,B}H^{DD}_{m,n}-\widetilde{H}^{DB}_{m,B}\widetilde{H}^{CD}_{k,m,n}\Gamma^{ C}_{k}\gamma^{D}_{n,m,k}\right)^{2}}-\frac{\lambda}{\ln 2\left(1+\gamma^{D}_{n,m,k} \right)}\;\;, \tag{19}\] \[=0,\forall k\in\mathcal{S}_{m}\] \[\lambda\left[\log_{2}(1+\Gamma^{D}_{m})-\sum\limits_{k\in \mathcal{S}_{m}}\log_{2}(1+\gamma^{D}_{n,m,k})\right]=0. \tag{20}\]  
From Equation (19), it can be easily found that the optimal \(\lambda\) must be positive, since \(\lambda=0\) implies that the equation \(\eta\sigma^{2}_{0}H^{CB}_{k,B}H^{DD}_{m,n}\left(H^{CB}_{k,B}+\widehat{H}^{CD}_{k,m,n}\Gamma^{C}_{k}\right)=0\) is satisfied for all \(k\in\mathcal{S}_{m}\), which is impossible in practice. With simple mathematical operations, Equation (19) can be rewritten into following quadratic equation:  
\[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]0  
where\[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]1 \[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]2 \[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]3 \[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]4  
Then we can easily get  
\[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]5  
Therefore, the available solution to Equation (21) can be obtained as follows:  
\[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]6  
From the above equation, we find  
\[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]7 \[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]8 \[\min_{\{\gamma^{D}_{n,m,k}\}\sum\limits_{k\in\mathcal{S}_{m}} \Phi\left(\gamma^{C\;*}_{k,m},\gamma^{D}_{n,m,k}\right)\]9  
which contradicts Equation (14). Consequently, the optimal solution to Equation (21) is given by  
\[s.t. C1:\sum\limits_{k\in\mathcal{S}_{m}}\log_{2}(1+\gamma^{D}_{n,m,k })\geq\log_{2}(1+\Gamma^{D}_{m}), \tag{16}\]0 \[s.t. C1:\sum\limits_{k\in\mathcal{S}_{m}}\log_{2}(1+\gamma^{D}_{n,m,k })\geq\log_{2}(1+\Gamma^{D}_{m}), \tag{16}\]1  
where \(\left[x\right]_{0}^{\gamma_{n,m,k,\max}^{D}}\) represents the projection onto the interval \(\left[0,\gamma_{n,m,k,\max}^{D}\right]\). Furthermore, according to Equation (20), the optimal \(\gamma_{n,m,k}^{D}\) must satisfy  
\[s.t. C1:\sum\limits_{k\in\mathcal{S}_{m}}\log_{2}(1+\gamma^{D}_{n,m,k })\geq\log_{2}(1+\Gamma^{D}_{m}), \tag{16}\]2  
Given that \(\gamma_{n,m,k}^{D}\) in Equation (27) depends on \(\lambda\) in a highly nonlinear way, directly computing the optimal \(\lambda\) is difficult. Nevertheless, the optimal \(\lambda\) can be efficiently found via bisection, for which one needs to know an interval containing the optimal \(\lambda\). The following result provides such an interval. **Lemma 1**.