## 3 Coordinated Multicast Transmission Design
Specifically, the problem in (13) is handled via iteratively solving the following sequence of optimization problems  
\[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]2 \[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]3 subject to \[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]4 \[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]5  
where the superscript \(i\) denotes the iteration index, \(\mathbf{\Lambda}^{\left(i\right)}=\left\{\mathbf{\Lambda}_{1}^{\left(i\right)},\cdots,\mathbf{\Lambda}_{U}^{\left(i\right)}\right\}\), and the gradient of \(\mathcal{R}_{k,u,2}\left(\mathbf{\Lambda}\right)\) with respect to \(\mathbf{\Lambda}_{\ell}\) for \(\ell\
eq u\) is a diagonal matrix, whose diagonal elements can be explicitly calculated as follows  
\[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]6  
We can observe that each subproblem in (18) is a convex program and can be more efficiently solved compared with the optimization problem in (13). In addition, from [35], it is not difficult to verify that the sequence \(\left\{\mathbf{\Lambda}^{\left(i\right)}\right\}_{i=0}^{\infty}\) generated by the proposed MM optimization approach for coordinated beam domain multi-cell massive MIMO multicast transmission in (18) monotonically converges to a stationary point of the problem in (13). The computational complexity of \(\mathcal{R}_{k,u,1}\left(\mathbf{\Lambda}\right)\) in (18) is still high due to the involved expectation operation. To further reduce the optimization complexity, we utilize the large dimensional random matrix theory [36] to calculate the deterministic equivalent (DE) of \(\mathcal{R}_{k,u,1}\left(\mathbf{\Lambda}\right)\) as follows  
\[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]7  
where \(\mathbf{\Gamma}_{k,u}\), \(\widetilde{\mathbf{\Gamma}}_{k,u}\), and \(\widetilde{\mathbf{\Phi}}_{k,u}\) can be obtained by solving the following fixed-point equations  
\[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]8 \[\left\{\mathbf{Q}_{1}^{\mathrm{op}},\cdots,\mathbf{Q}_{U}^{\mathrm{op}} \right\}=\underset{\mathbf{Q}_{1},\cdots,\mathbf{Q}_{U}}{\arg\max} \sum_{u=1}^{U}\mathcal{R}_{u}\]9 subject to \[\mathrm{subject\ to} \mathrm{tr}\left(\mathbf{Q}_{u}\right)\leq P_{u},\]0 \[\mathrm{subject\ to} \mathrm{tr}\left(\mathbf{Q}_{u}\right)\leq P_{u},\]1  
For the sequence of optimization problems in (24), the DE result \(\overline{\mathcal{R}}_{k,u,1}\left(\mathbf{\Lambda}\right)\) is concave with respect to \(\mathbf{\Lambda}\). In addition, the DE result \(\overline{\mathcal{R}}_{k,u,1}\left(\mathbf{\Lambda}\right)\) is a quite good approximation of \(\mathcal{R}_{k,u,1}\left(\mathbf{\Lambda}\right)\) for massive MIMO transmission [36]. Thus, the sequence of optimization problems in (24) can be efficiently solved. Based on the above formulation, we propose the coordinated beam domain power allocation algorithm for multi-cell massive MIMO multicast transmission in Algorithm 1.