## 3 Proposed Optimization Algorithm
### BS Transmit Power Allocation
#### 3.1.2 Two-Layer Approach for Solving Problem (P1)
\[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]0 \[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]1 (53) \[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]2 (54) \[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]3 (55) \[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]4 (56) \[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]5 (57) \[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]6 (58) \[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]7

The application of the reformulation-linearization methods described above reveals the mapping between the solutions of (38) and (51). It is worth noting that the transformed MILP problem (P2\({}^{\prime}\)) maintains the same optimal \(\mathbf{\Gamma}\) as the original problem (P2). This is because the reformulation-linearization process involves only variable substitution and exact linearization. This ensures that problem (P2\({}^{\prime}\)) can be effectively solved using MILP solvers like CPLEX or Gurobi. Consequently, we can employ a two-layer iterative algorithm to solve problem (P2). The complete algorithm is outlined in Algorithm 2. ### Overall Algorithm Description

In this subsection, we describe the whole algorithm that solves the weighted sum-SINR maximization problem (P0) iteratively. The BS transmit power \(\mathbf{P}\) is first solved by Algorithm 1 with fixed BS selections. Based on the obtained BS power, the BS selection \(\mathbf{\Gamma}\) is optimized by Algorithm 2. If the objective function values of two consecutive iterations differ by less than a predefined threshold \(\varepsilon\), i.e., \(F^{i}-F^{i-1}\leq\varepsilon\) (\(F\) represents the value of objective function), we consider that the entire algorithm has converged. In light of the above description, the main procedures of the overall algorithm are outlined in Algorithm 3. ```
1:Initialize the BS selections \(\mathbf{\Gamma}^{0}\) that meets conditions (12) and (13), and set iteration index \(i=1\). And \(\varepsilon>0\) is the update accuracy control factor. 2:loop
3:Given the BS selections \(\mathbf{\Gamma}^{i-1}\), solve the power allocation problem (P1) and obtain the solution \(\mathbf{P}^{i}\). 4:Given the BS power allocation \(\mathbf{P}^{i}\), solve the BS selection problem (P2) and obtain the solution \(\mathbf{P}^{i}\). 5:Update \(F^{i}=\sum\limits_{n\in\mathcal{N}}\frac{U_{n}(\mathbf{P}^{i},\mathbf{\Gamma} ^{i})}{D_{n}(\mathbf{P}^{i},\mathbf{\Gamma}^{i})}\). 6:if\(F^{i}-F^{i-1}\leq\varepsilon\)then
7:break
8:endif
9:endloop
10:\(\mathbf{P}^{*}=\mathbf{P}^{i},\mathbf{\Gamma}^{*}=\mathbf{\Gamma}^{i}\). ```

**Algorithm 3** The Whole AlgorithmFurthermore, we analyze the convergence of the whole algorithm. By performing Algorithm 1, a better power allocation \(\mathbf{P}^{i}\) is achieved with a fixed BS selection \(\mathbf{\Gamma}^{i-1}\) in the \(i\)-th iteration. Hence, we have

\[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]8

Similarly, given the power allocation \(\mathbf{P}^{i}\), by Algorithm 2, the BS selection is designed to maximize the weighted sum-SINR. Thus, we can obtain

\[v_{3}=\sum_{n\in\mathcal{N}}\mu_{n}\rho_{n}\sigma^{2}. \tag{32}\]9

Based on (60) and (61), we obtain

\[F=(v_{1}-v_{2})P_{m,n}-v_{3}. \tag{33}\]0

The analysis presented above demonstrates that the objective value of the original problem (P0) consistently exhibits non-decreasing behavior after each iteration. Consequently, the convergence of the algorithm is guaranteed. The non-decreasing behavior of the objective value suggests that the algorithm is effectively moving towards a better solution with each iteration, ultimately converging to a satisfactory solution. This conclusion is further supported by the theoretical foundations of the algorithm, which ensure its convergence to a locally optimal solution in a finite number of iterations. ## 4 Numerical Results

In this section, we evaluate the performance of our proposed algorithm.