## 4 Maximization of Weighted Sum Rate Based on ZF Scheme
### Algorithm Summary and Analysis  
Algorithm 1 summarizes the process for a given distribution of all \(\gamma_{k,m}\)s, i.e., \(\{\hat{\mathcal{U}}_{t},\hat{\mathcal{V}}_{t}\}\). Steps 2 to 5 iteratively update to solve convex optimization Problem **P4**. Upon specifying the range of possibilities, this algorithm eventually converges to a solution satisfying the KKT conditions of optimization Problem **P4**[34]. Specifically, the application of the SCA method requires meeting three conditions for achieving convergence. In Problem **P4**, these conditions are as follows: The term \(\mathcal{F}(\gamma_{k,m}),\forall\gamma_{k,m}\in\mathcal{U}\) in the objective of (24) is non-convex, and we replace it with its first-order Taylor-series expansion \(\mathcal{F}(\gamma_{k,m}^{(t-1)})+\mathcal{F}^{\prime}(\gamma_{k,m}^{(t-1)})( \gamma_{k,m}-\gamma_{k,m}^{(t-1)})\). At the local point \(\gamma_{k,m}^{(t-1)}\), we ensure \(\mathcal{F}(\gamma_{k,m})\leq\mathcal{F}(\gamma_{k,m}^{(t-1)})+\mathcal{F}^{ \prime}(\gamma_{k,m}^{(t-1)})(\gamma_{k,m}-\gamma_{k,m}^{(t-1)})\), which fulfills condition 1. Furthermore, the values before and after substitution at the local point \(\gamma_{k,m}^{(t-1)}\) are equal, both being \(\mathcal{F}(\gamma_{k,m}^{(t-1)})\), satisfying condition 2. Finally, their gradients at the local point \(\gamma_{k,m}^{(t-1)}\) are also consistent, meeting condition 3. Besides these conditions, Problem **P4** meets the requirements of convex optimization and satisfies [PERSON]'s condition in each iteration of the SCA process. According to ([34], Theorem 1), we can prove that the result obtained by Algorithm 1 is a KKT stationary point. Moreover, if this point lies within the feasible domain, it further indicates a local optimal point. By analyzing all \(\{\hat{\mathcal{U}}_{t},\hat{\mathcal{V}}_{t}\}\), we can obtain the optimal solution across the entire feasible domain. ```
Input:\(\{\hat{\mathcal{U}}_{t},\hat{\mathcal{V}}_{t}\}\), \(\tau,P_{s},\epsilon_{\text{SCA}},T_{\text{max}},\mathbf{d},\mathbf{g}_{k},\mu_{k},\forall k\) Output:\(W\), and \(\widetilde{R}_{all}\) Initialize:\(W^{(0)},\lambda_{k,1}^{(0)},\lambda_{k,2}^{(0)},\gamma_{k,1}^{(0)},\gamma_{k,2}^{(0)}, \theta_{k,1}^{(0)},\theta_{k,2}^{(0)},\forall k\) and the iteration index \(t=0\); while\(\sum_{\forall k}\left[\left(\mathbf{w}_{k}^{(t)}-\mathbf{w}_{k}^{(t-1)}\right)^{2}+ \sum_{\forall m}\left(\lambda_{k,m}^{(t)}-\lambda_{k,m}^{(t-1)}\right)^{2}+ \sum_{\forall m}\left(\gamma_{k,m}^{(t)}-\gamma_{k,m}^{(t-1)}\right)^{2}\right]> \mathcal{E}_{SCA}\) or \(t<T_{\text{max}}\)do

Use the toolbox to solve Problem **P4**;

\(t\gets t+1\);

\(\theta_{k,1}^{(t-1)}=\frac{\sqrt{\lambda_{k,1}^{(t-1)}}}{\gamma_{k,1}^{(t-1)}}\) and \(\theta_{k,2}^{(t-1)}=\frac{\sqrt{\lambda_{k,2}^{(t-1)}}}{\gamma_{k,2}^{(t-1)}}, \forall k\). ```  
**Algorithm 2**The proposed algorithm for Problem **P1**. Algorithm 1, in Step 4, we add a feasibility check. This is because, regardless of whether \(n_{s}\mathbf{g}_{k}^{T}w_{k}\) is greater than 0, the inequality \(\frac{n_{s}\mathbf{g}_{k}^{T}w_{k}}{\sqrt{\lambda_{k,1}}}\geq\frac{n_{s}\mathbf{g}_{k} ^{T}w_{k}}{\sqrt{\lambda_{k,2}}},\forall k\) always exists. By preliminarily assessing feasibility, we can reduce the possibilities for discussion and decrease the likelihood of the program entering an infinite loop. ```
Input:\(\tau,P_{s},\epsilon_{\text{SCA}},T_{\text{max}},\mathbf{d},\mathbf{g}_{k},\mu_{k}, \forall k\) Output:\(W\) and \(\widetilde{R}_{all}\) Initialize: List the possible scenarios \(\{\hat{\mathcal{U}}_{t},\hat{\mathcal{V}}_{t}\},\forall t=1,\cdots,2^{2K}\); for\(t=1,\cdots,2^{2K}\)do if\(\frac{n_{s}\mathbf{g}_{k}^{T}w_{k}}{\sqrt{\lambda_{k,1}}}=\frac{n_{s}\mathbf{g}_{k}^{T}w_{k}}{ \sqrt{\lambda_{k,2}}},\forall k\)then  running Algorithm 1 with \(\{\hat{\mathcal{U}}_{t},\hat{\mathcal{V}}_{t}\}\) to obtain \(\widetilde{R}_{all}(t)\) and \(\mathbf{W}(t)\); else go back to step 2;
```  
**Algorithm 3**The proposed algorithm for Problem **P1**.