## 3 The Proposed Optimization Algorithm
### Power Allocation  
With the fixed receive beamforming vector \(\mathbf{v}_{m,k}\) and UE association indicator \(\alpha_{m,k}\), problem (7) is reduced to the following form:  
\[ \begin{split}\max_{p_{m,k}}&\sum_{m\in\mathcal{R}} \sum_{k\in\mathcal{U}_{m}}R_{m,k}\\ \text{s.t.}&\text{C1}^{\prime}:p_{m,k}\leq p_{k}^{ max},\quad\forall m\in\mathcal{R},\forall k\in\mathcal{U}_{m},\\ &\text{C4}^{\prime}:\sum_{k\in\mathcal{U}_{m}}R_{m,k}\leq C_{m}, \ \forall m\in\mathcal{M},\\ &\text{C5}^{\prime}:R_{m,k}\geq r_{k},\ \forall m\in\mathcal{R}, \forall k\in\mathcal{U}_{m},\end{split}  \tag{11}\]  
By introducing the Lagrange multiplier \(\rho_{k}\) associated with the constraint C5\({}^{\prime}\), the partial Lagrange function of problem (11) can be derived as  
\[ \begin{split}\max_{p_{m,k}}&\sum_{m\in\mathcal{R}} \sum_{k\in\mathcal{U}_{m}}R_{m,k}+\sum_{m\in\mathcal{R}}\sum_{k\in\mathcal{U}_ {m}}\rho_{k}(R_{m,k}-r_{k})\\ \text{s.t.}&\text{C1}^{\prime},\text{C4}^{\prime} \end{split}  \tag{12}\]  
To tackle the non-convexity of (12), we first introduce the quadratic transform method in [20] to convert log-fractional rate functions in the objective function into convex quadratic expressions. By introducing auxiliary variables \(\gamma_{m,k}\) and \(\eta_{m,k}\), the log-fractional rate function can be transformed into  
\[ \begin{split}\tilde{R}_{m,k}\triangleq&\log(1+\gamma _{m,k})-\gamma_{m,k}+2\eta_{m,k}\sqrt{(1+\gamma_{m,k})\left|\mathbf{v}_{m,k} \mathbf{h}_{m,k}\right|^{2}p_{m,k}}\\ &-\eta_{m,k}^{2}\left(\sum\limits_{j\in\{\mathcal{U}_{m}\backslash \mathcal{S}_{m,k}\}\cup k}\left|\mathbf{v}_{m,k}\mathbf{h}_{mj}\right|^{2}p_{m,j}+\sum\limits_{i\in\mathcal{R},i\
eq m}\sum\limits_{j\in\mathcal{U}_{k}} \left|\mathbf{v}_{m,k}\mathbf{h}_{ij}\right|^{2}p_{ij}+\left\|\mathbf{v}_{m,k} \right\|^{2}\sigma_{m}^{2}\right)\end{split}  \tag{13}\]  
By substituting \(\tilde{R}_{m,k}\) into (12), the problem is then transformed into the following:  
\[ \begin{split}\max_{p_{m,k}}&\sum_{m\in\mathcal{R}} \sum_{k\in\mathcal{U}_{m}}\tilde{R}_{m,k}+\sum_{m\in\mathcal{R}}\sum_{k\in \mathcal{U}_{m}}\rho_{k}\big{(}\tilde{R}_{m,k}-r_{k}\big{)}\\ \text{s.t.}&\text{C1}^{\prime},\text{C4}^{\prime} \end{split}  \tag{14}\]  
Given \(p_{m,k}\), \(\gamma_{m,k}\) and \(\eta_{m,k}\) can be updated by  
\[ \begin{split}\gamma_{m,k}=\frac{\left|\mathbf{v}_{m,k}\mathbf{h}_ {m,k}\right|^{2}p_{m,k}}{\sum\limits_{j\in\mathcal{U}_{m}\backslash\mathcal{S} _{m,k}}\left|\mathbf{v}_{m,k}\mathbf{h}_{mj}\right|^{2}p_{m,j}+\sum\limits_{i\in \mathcal{R},i\
eq m}\sum\limits_{j\in\mathcal{U}_{i}}\left|\mathbf{v}_{m,k} \mathbf{h}_{ij}\right|^{2}p_{i,j}+\left\|\mathbf{v}_{m,k}\right\|^{2}\sigma_{m }^{2}}\end{split}  \tag{15}\]  
\[ \begin{split}\eta_{m,k}=\frac{\sqrt{(1+\gamma_{m,k})\left|\mathbf{ v}_{m,k}\mathbf{h}_{m,k}\right|^{2}p_{m,k}}}{\sum\limits_{j\in\{\mathcal{U}_{m} \backslash\mathcal{S}_{m,k}\}\cup k}\left|\mathbf{v}_{m,k}\mathbf{h}_{mj} \right|^{2}p_{m,j}+\sum\limits_{i\in\mathcal{R},i\
eq m}\sum\limits_{j\in \mathcal{U}_{i}}\left|\mathbf{v}_{m,k}\mathbf{h}_{ij}\right|^{2}p_{i,j}+\left\| \mathbf{v}_{m,k}\right\|^{2}\sigma_{m}^{2}}\end{split}  \tag{16}\]  
With fixed \(\gamma_{m,k}\) and \(\eta_{m,k}\), the residual power allocation of (14) is convex without C4\({}^{\prime}\). Accordingly, during the iterative process, we temporarily disregard constraint C4\({}^{\prime}\). Following the completion of the iterations, adjustments will be made to the power allocation in order to satisfy constraint C4\({}^{\prime}\). The corresponding Lagrangian dual function is derived as  
\[ \begin{split}\mathcal{L}(p_{m,k})=\sum_{m\in\mathcal{R}}\sum_{k \in\mathcal{U}_{m}}\big{(}\tilde{R}_{m,k}+\rho_{k}\big{(}\tilde{R}_{m,k}-r_{k} \big{)}\big{)}\end{split}  \tag{17}\]  
The Karush-Kuhn-Tucker (KKT) conditions are\[