## 3 Proposed Optimization Algorithm
### BS Transmit Power Allocation
#### 3.1.2 Two-Layer Approach for Solving Problem (P1)
The inner layer of this approach involves solving the parameterized problem for given auxiliary parameters \((\mathbf{\mu}^{\prime},\mathbf{\rho}^{\prime})\succeq 0\), which is expressed as

\[\max_{\mathbf{\Gamma}} \sum_{n\in\mathcal{N}}\mu_{n}^{\prime}\left[a_{n}\sum_{m\in \mathcal{M}}\gamma_{m,n}d_{m,n}P_{m,n}-\rho_{n}^{\prime}\left(\sigma^{2}+\sum \limits_{j\in\mathcal{N},j\
eq n}\frac{\sum\limits_{m\in\mathcal{M}}\gamma_{m,j}P_{m,j}u_{m,n}d_{m,j}}{\sum\limits_{m\in\mathcal{M}}\gamma_{m,j}d_{m,j}} \right)\right]\] (38) s.t. (19)-(21). The outer layer is similar to that of Algorithm 1, i.e., update \((\mathbf{\mu}^{\prime},\mathbf{\rho}^{\prime})\), via the modified Newton method, (36) and (37). Due to the similarity, we left the detailed implementation of the outer layer in Algorithm 2 and focus on the discussion of solving the inner-layer problem (38). ```
1:Input P and the update accuracy control factor \(\epsilon>0\). 2:Initialize \((\mathbf{\mu}^{\prime},\mathbf{\rho}^{\prime})\succeq 0\). 3:loop
4: Given \((\mathbf{\mu}^{\prime},\mathbf{\rho}^{\prime})\), obtain the optimal \(\mathbf{\Gamma}\) to problem (P2\({}^{\prime}\)). 5: Update \((\mathbf{\mu}^{\prime},\mathbf{\rho}^{\prime})\) using (36) and (37). 6:if\(\sum_{n\in\mathcal{N}}(|\psi_{n}^{1}|^{2}+|\psi_{n}^{2}|^{2})<\epsilon\)then
7:break
8:endif
9:endloop
10:Output:\(\mathbf{\Gamma}\). ```

**Algorithm 2** Two-Layer Iterative Algorithm for Problem (P2)

It is noted that (38) is still an integer linear fractional programming (ILFP) problem with respect to \(\mathbf{\Gamma}\) because of the fractional component \(\sum\limits_{m\in\mathcal{M}}\gamma_{m,j}P_{m,j}u_{m,n}d_{m,j}\bigg{/}\sum \limits_{m\in\mathcal{M}}\gamma_{m,j}d_{m,j}\) in the objective function. To proceed, we utilize the reformulation-linearization technique [42] to recast the ILFP problem into an equivalent MILP problem. This transformation enables us to optimize the problem efficiently using standard MILP solvers. The reformulation-linearization approach consists of two main stages. In the initial stage, we aim to reframe the fractional component in the objective function (38) into an equivalent bilinear term. To achieve this, we introduce supplementary continuous variables and impose equality constraints. Subsequently, in the second stage, we aim to linearize the bilinear term appearing in both the objective function and constraints. The specific steps involved in this transformation are outlined below. Reformulation: To reformulate the problem, we introduce a new variable \(z_{j}>0\) defined as

\[z_{j}=\frac{1}{\sum\limits_{m\in\mathcal{M}}\gamma_{m,j}d_{m,j}}. \tag{39}\]

It is important to note that this variable \(z_{j}\) is positive. By substituting (39) into the objective function (38), we can successfully reformulate the problem into the following equivalent form:

\[v_{2}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\left(\sum_{n\in \mathcal{N}}\frac{\sum\limits_{m\in\mathcal{M}}\gamma_{m,n}u_{m,n}d_{m,n}}{ \sum\limits_{m\in\mathcal{M}}\gamma_{m,n}d_{m,n}}-\frac{\sum\limits_{m\in \mathcal{M}}\gamma_{m,n}u_{m,n}d_{m,n}}{\sum\limits_{m\in\mathcal{M}}\gamma_{ m,n}d_{m,n}}\right),\]0 (40) s.t. (19)-(21) \[v_{2}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\left(\sum_{n\in \mathcal{N}}\frac{\sum\limits_{m\in\mathcal{M}}\gamma_{m,n}u_{m,n}d_{m,n}}{ \sum\limits_{m\in\mathcal{M}}\gamma_{m,n}d_{m,n}}-\frac{\sum\limits_{m\in \mathcal{M}}\gamma_{m,n}u_{m,n}d_{m,n}}{\sum\limits_{m\in\mathcal{M}}\gamma_{ m,n}d_{m,n}}\right),\]1

where the equality constraint (42) is derived from (39) to define the variable \(z_{j}\). This reformulation allows us to manipulate the problem in a more tractable manner, paving the way for further optimization techniques. It is worth mentioning that the binary variables \(\gamma_{m,j}\) remain unchanged in the previous step, leading to a new bilinear term \(\gamma_{m,j}z_{j}\) emerging in the objective function (40) as a result of the reformulation.