## 3 Proposed Approach for Addressing Problem (5)
### Deep Learning-Based Scheme for Optimizing \(\alpha_{k}\)
\[\left(\sum_{i=1}^{K}\text{Tr}(\mathbf{Q}_{k,PS}\mathbf{P}_{i})+\sum_{ g=1}^{G}\text{Tr}\big{(}\mathbf{H}_{k,PS}\mathbf{E}_{g}\big{)}+\sigma_{k}^{2}+ \frac{\gamma_{k}^{2}}{\theta_{k}}\right)\varphi-\text{Tr}(\mathbf{Q}_{k,PS} \mathbf{P}_{0})\leq 0,\,\forall k \tag{11c}\] \[\frac{\epsilon_{k}^{PS}}{\varsigma_{k}^{PS}(1-\theta_{k})}-\sum_{ i=0}^{K}\text{Tr}(\mathbf{Q}_{k,PS}\mathbf{P}_{i})-\sum_{g=1}^{G}\text{Tr} \big{(}\mathbf{H}_{k,PS}\mathbf{E}_{g}\big{)}-\sigma_{k}^{2}\leq 0,\, \forall k\] (11d) \[-\epsilon_{g}^{EH}\sum_{i=0}^{K}\text{Tr}\big{(}\mathbf{Q}_{g,EH} \mathbf{P}_{i}\big{)}-\epsilon_{g}^{EH}\sum_{j=1}^{G}\text{Tr}\big{(}\mathbf{ H}_{g,EH}\mathbf{E}_{j}\big{)}+\epsilon_{g}^{EH}\leq 0,\,\forall g\] (11e) \[\mathbf{P}_{0},\mathbf{P}_{k},\mathbf{E}_{g}\succeq 0,\,\forall k,\,\forall g\] (11f) \[0<\theta_{k}<1,\,\forall k. \tag{11g}\]  
Problem (11) is convex and can be solved efficiently using the CVX toolbox in MATLAB [39]. Problem (11) involves \(K+1\) matrices of size \(M\times M\) and \(G\) matrices of size \(N\times N\), along with the \(L_{C}=3K+G\) linear constraint variables. Therefore, the computational complexity of addressing problem (11) amounts to \(\mathcal{O}\Big{(}\sqrt{(K+1)M+GN}\Big{(}(K+1)^{3}M^{6}\)\(+\)\(G^{3}N^{6}+L_{c}\big{(}(K+1)M^{2}+GN^{2}\big{)}\big{)}\log(1/\zeta)\big{)}\) while maintaining a solution accuracy of \(\zeta>0\)[40]. The optimal solutions are represented in problem (11) as \(\Big{\{}\mathbf{P}_{k^{\prime}}^{*},\mathbf{E}_{g}^{*}\Big{\}}\). When the matrix solutions have a rank of one, the optimal beamforming vectors, \(\Big{\{}\mathbf{P}_{i}^{*},\mathbf{e}_{g}^{*}\Big{\}}\), are given by  
\[\mathbf{p}_{i} =\sqrt{\lambda_{H}(\mathbf{P}_{i})}\mathbf{v}_{H,\mathbf{P}_{i^{ \prime}}},\,i=0,\ldots,K \tag{12a}\] \[\mathbf{e}_{g} =\sqrt{\lambda_{H}\big{(}\mathbf{E}_{g}\big{)}}\mathbf{v}_{H, \mathbf{E}_{g^{\prime}}},\,g=1,\ldots,G, \tag{12b}\]  
where \(\lambda_{H}(\mathbf{A})\) represents the largest eigenvalue of the matrix \(\mathbf{A}\), and \(\mathbf{v}_{H,\mathbf{A}}\) denotes its corresponding eigenvector. If \(\Big{\{}\mathbf{P}_{i}^{*},\mathbf{E}_{g}^{*}\Big{\}}\) are rank-one, alternative approaches, such as the penalty function method [41] or the Gaussian randomization technique [20], can be employed to approximate the beamforming vectors. The following details the penalty function method for the scenario where the matrix variables \(\Big{\{}\mathbf{P}_{i}^{*},\mathbf{E}_{g}^{*}\Big{\}}\) are not rank-one. The proposed penalty-based method is based on the definition of \(\Big{\{}\mathbf{P}_{i}^{*},\mathbf{E}_{g}^{*}\Big{\}}\) being positive semidefinite matrices, which satisfy the conditions \(\text{Tr}(\mathbf{P}_{i})\geq\lambda_{H}(\mathbf{P}_{i})\) and \(\text{Tr}\big{(}\mathbf{E}_{g}\big{)}\geq\lambda_{H}(\mathbf{E}_{g})\), where the matrices \(\Big{\{}\mathbf{P}_{i}^{*},\mathbf{E}_{g}^{*}\Big{\}}\) are rank-one if \(\text{Tr}(\mathbf{P}_{i})=\lambda_{H}(\mathbf{P}_{i})\) and \(\text{Tr}\big{(}\mathbf{E}_{g}\big{)}=\lambda_{H}(\mathbf{E}_{g})\). Hence, the proposed penalty-based approach aims to minimize \(\text{Tr}(\mathbf{P}_{i})-\lambda_{H}(\mathbf{P}_{i})\) and \(\text{Tr}\big{(}\mathbf{E}_{g}\big{)}-\lambda_{H}\big{(}\mathbf{E}_{g}\big{)}\) by introducing a penalty factor, \(\kappa\), and incorporating these terms into the objective function as follows:  
\[ \begin{array}{rl}\min_{\mathbf{P}_{0},\big{\{}\mathbf{P}_{1}, \mathbf{E}_{g}\big{\}}}&\sum\limits_{i=0}^{K}\text{Tr}(\mathbf{P}_{i})+\sum \limits_{g=1}^{G}\text{Tr}(\mathbf{E}_{g})+\kappa\sum\limits_{i=0}^{K}(\text{Tr} (\mathbf{P}_{i})-\lambda_{H}(\mathbf{P}_{i}))+\kappa\sum\limits_{g=1}^{G} \big{(}\text{Tr}\big{(}\mathbf{E}_{g}\big{)}-\lambda_{H}(\mathbf{E}_{g})\big{)} \\ &\text{subject to \eqref{eq:11b},\ldots,\eqref{eq:11g}.}\end{array}  \tag{13}\]  
Problem (13) is non-convex because of the terms \(-\lambda_{H}(\mathbf{P}_{i})\) and \(-\lambda_{H}\big{(}\mathbf{E}_{g}\big{)}\).