# Energy-Efficient Resource and Power Allocation for Underlay Multicast Device-to-Device Transmission
## 4 Proposed Heuristic Energy-Efficient Resource and Power Allocation Algorithm  
According to (12), we can deduce that in order to improve energy-efficiency of all the D2D multicast clusters, we have to either decrease the transmit power or to increase the aggregate transmission rate. However, due to the fraction form of the objective function built in (12), it is a non-convex optimization problem. Hence, it is difficult to obtain the optimal solution directly. In the following part, we will first solve the overall energy-efficiency optimization problem by adopting a heuristic resource allocation strategy. Firstly, we consider how to increase the numerator of expression (12). From the resource assignment point of view, in order to improve the aggregate D2D throughput, higher values of SINR are desirable. From constraints (12a)-(12f), we can infer that a smaller value of \(G_{n,m}^{C2D}\) means less interference from co-channel CU \(n\) to D2D cluster \(m\), which will result in higher \(\gamma_{m}^{D2D}\) and D2D throughput. Hence, the fundamental idea of the proposed scheme is to pick up the CU who generates less interference to the co-channel D2D cluster. To achieve that target, we build a channel state information (CSI) matrix \(\mathbf{G_{M\times N}^{C2D}}\), which is composed of estimated channel gain information of each D2D cluster from the interfering CU respectively, where \(\mathbf{G_{M\times N}^{C2D}}\) is expressed as  
\[\mathbf{G_{M\times N}^{C2D}}=\left[ \begin{array}{c}G_{1,1}^{C2D},G_{21}^{C2D},...,G_{N,1}^{C2D}\\ G_{1,2}^{C2D},G_{2,2}^{C2D},...,G_{N,2}^{C2D}\\ \vdots\\ G_{1,M}^{C2D},G_{2,M}^{C2D},...,G_{N,M}^{C2D}\end{array} \right] \tag{13}\]  
In the matrix \(\mathbf{G_{M\times N}^{C2D}}\), each element stands for the CSI value from the co-channel CU to the corresponding D2D multicast cluster. We assume that the CSI between each CU and each D2D cluster can be obtained by each cluster head individually. Actually, such information can be initially obtained through information change between each cluster head and D2D receivers. Then, the CSI information between each cluster and each CU can be gathered at the BS side through control information exchange. Consequently, we can find the minimum \(G_{n,m}^{C2D}\), \(n\in\mathcal{N},m\in\mathcal{M}\) in each row so as to pair up the CU which brings the least interference to the D2D multicast cluster. By doing the same procedure for each D2D multicast cluster, the optimal resource assignment between CU and the corresponding D2D multicast cluster can be decided. After picking out the cellular resource for each D2D cluster, the next step is to decrease the denominator of expression (12). This is interpreted as to determine the minimum transmit power of each CU and D2D cluster respectively, which also satisfies the minimum SINR threshold requirement. Suppose a CU \(n\) shares RB resource with a D2D cluster \(m\) after resource pairing process. By using the Equations (8) and (9) and substituting \(\gamma_{n}^{CU}\) and \(\gamma_{m}^{D2D}\) from expression (3) and (4) respectively, we can determine the optimal transmit power of a D2D cluster as the network can guarantee reliable transmission as follows  
\[\left\{\begin{array}{c}P_{m}^{D2D_{m}}=\max_{\forall m\in\mathcal{M},k\in \mathcal{K}_{m,m},m\in\mathcal{N}}\frac{\Gamma_{m}^{D2D}\Gamma_{m}^{CU}G_{n,k}^ {C2D}\sigma_{n}^{2}+\Gamma_{m}^{D2D}G_{n,k}^{C}\sigma_{n}^{2}}{\Gamma_{m,k}^{ C2  
In the proposed scheme, in order to maximize the overall energy-efficiency of D2D multicast clusters, we first search for the optimal RB assignment between CUs and D2D clusters. This leads to increased total data rates of D2D multicast transmission. Then, combined with the minimum SINR requirement, we find out the minimum transmit power of both CUs and D2D clusters to ensure reliable D2D multicast transmission. Hence, the maximization problem built in expression (12) is solved by a two-step way, where the computational complexity of the proposed strategy is \((MN)+(M\times f|\mathcal{K}_{m}|)\).