## 3 Solution of the Optimization Problem
### Optimality Conditions for SDP Relaxation
: The Lagrangian dual function of problem (20) can be modeled as \[\mathfrak{L}(\mathbf{W},\mathbf{V},\mathbf{V}_{0},\mathbf{X}_{k},\mathbf{Y}_{k},\alpha, \beta,\gamma,\zeta_{k},\mathbf{\Theta},\mathbf{\Xi},\mathbf{\Omega},\rho,t,\tau_{E,k},\bar{ \lambda}_{k},\bar{\mu}_{k})\] \[=t+\sum_{k=1}^{K_{E}}\text{Tr}\big{(}\mathbf{X}_{k}N_{E,k}\big{(}\mathbf{W },\mathbf{V}_{0},\bar{\lambda}_{k}\big{)}\big{)}\] \[+\alpha\big{[}P-\text{Tr}\big{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_{0}\big{)} \big{]}\] \[+\beta\big{[}\text{Tr}\big{(}\big{(}\mathbf{W}-r\mathbf{V}-r\mathbf{V}_{0} \big{)}\mathbf{H}\big{)}-\frac{\sigma_{I}^{2}r}{\rho}\big{]}\] \[+\gamma\big{[}\text{Tr}\big{(}\big{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_{0} \big{)}\mathbf{H}\big{)}-\frac{\omega_{I}}{1-\rho}\big{]} \tag{27}\] \[\mathbf{W}^{*}=\sum_{n=1}^{N_{T}-p}b_{n}\mathbf{\pi}_{1,n}\mathbf{\pi}_{1,n}^{H}+a\mathbf{\iota }^{H}, \tag{23}\]0 \[\mathbf{W}^{*}=\sum_{n=1}^{N_{T}-p}b_{n}\mathbf{\pi}_{1,n}\mathbf{\pi}_{1,n}^{H}+a\mathbf{\iota }^{H}, \tag{23}\]1 \[\mathbf{W}^{*}=\sum_{n=1}^{N_{T}-p}b_{n}\mathbf{\pi}_{1,n}\mathbf{\pi}_{1,n}^{H}+a\mathbf{\iota }^{H}, \tag{23}\]2  
where \(\mathbf{X}_{k}\succeq\mathbf{0}\), \(\forall k\), \(\mathbf{Y}_{k}\succeq\mathbf{0}\), \(\forall k\), \(\alpha\geq 0\), \(\beta\geq 0\), \(\gamma\geq 0\), \(\zeta_{k}\geq 0\), \(\forall k\) are the dual variables for constraints C2, C9, C4, C1, C3, and C5, respectively. \(\mathbf{\Theta}\in\mathbb{H}_{+}^{\mathbb{N}}\), \(\mathbf{\Xi}\in\mathbb{H}_{+}^{\mathbb{N}}\) and \(\mathbf{\Omega}\in\mathbb{H}_{+}^{N}\) are the dual variables with respect to parameters \(\mathbf{W}\), \(\mathbf{V}\) and \(\mathbf{V}_{0}\), respectively. Let \(\tilde{\mathbf{G}}_{k}= \begin{bmatrix}\mathbf{I}_{N_{T}N_{R}}&\tilde{\mathbf{g}}_{k}\end{bmatrix} \), \(\mathbf{\Lambda}_{E,k}= \begin{bmatrix}\bar{\mu}_{k}I_{N_{T}N_{R}}&\mathbf{0}\\ \mathbf{0}&-v_{k}^{2}\bar{\mu}_{k}-\tau_{E,k}\end{bmatrix} \), \(\mathbf{\Lambda}_{E,k}= \begin{bmatrix}\bar{\lambda}_{k}I_{N_{T}N_{R}}&\mathbf{0}\\ \mathbf{0}&-v_{k}^{2}\bar{\lambda}_{k}+r_{k}\sigma_{E}^{2}\end{bmatrix} \), we can rewrite the \(N_{E,k}\big{(}\mathbf{W},\mathbf{V}_{0},\bar{\lambda}_{k}\big{)}\) and \(T_{E,k}\big{(}\mathbf{W},\mathbf{V}_{0},\bar{\mu}_{k},t\big{)}\) as follows:  
\[ \begin{split}& N_{E,k}\big{(}\mathbf{W},\mathbf{V}_{0},\bar{\lambda}_{k }\big{)}=\tilde{\mathbf{G}}_{k}^{H}\big{(}r_{k}\tilde{\mathbf{V}}_{0}-\tilde{\mathbf{W}} \big{)}\tilde{\mathbf{G}}_{k}+\tilde{\mathbf{\Lambda}}_{E,k},\\ & T_{E,k}\big{(}\mathbf{W},\mathbf{V}_{0},\mathbf{V},\bar{\mu}_{k},t\big{)}= \tilde{\mathbf{G}}_{k}^{H}\big{(}\tilde{\mathbf{W}}+\tilde{\mathbf{V}}+\tilde{\mathbf{V}}_{0} \big{)}\tilde{\mathbf{G}}_{k}+\mathbf{\Lambda}_{E,k}.\end{split}  \tag{28}\]  
For the convenience of analyzing the construction of the resulting solution, we must write back the variable \(\tilde{\mathbf{W}}\), \(\tilde{\mathbf{V}}\), and \(\tilde{\mathbf{V}}_{0}\) into their primal styles. Hence, we have  
\[\tilde{\mathbf{G}}_{k}^{H}\mathbf{X}_{k}\tilde{\mathbf{G}}_{k}= \begin{bmatrix}\mathbf{K}_{E,k}^ {(1,1)}&\mathbf{K}_{E,k}^{(1,2)}&\cdots&\mathbf{K}_{E,k}^{(1,N_{R})}\\ \mathbf{K}_{E,k}^{(2,1)}&\mathbf{K}_{E,k}^{(2,2)}&\cdots&\mathbf{K}_{E,k}^{(1,N_{R})}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{K}_{E,k}^{(N_{R},1)}&\mathbf{K}_{E,k}^{(N_{R},2)}&\cdots&\mathbf{K}_{E,k}^{(N_{R},N_{R})} \end{bmatrix} \in\mathbb{H}_{+}^{N_{T}N_{R}},\\ \tilde{\mathbf{G}}_{k}^{H}\mathbf{Y}_{k}\tilde{\mathbf{G}}_{k}= \begin{bmatrix}\mathbf{I}_{E,k}^ {(1,1)}&\mathbf{I}_{E,k}^{(1,2)}&\cdots&\mathbf{I}_{E,k}^{(1,N_{R})}\\ \mathbf{I}_{E,k}^{(2,1)}&\mathbf{I}_{E,k}^{(2,2)}&\cdots&\mathbf{I}_{E,k}^{(1,N_{R})}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{I}_{E,k}^{(N_{R},1)}&\mathbf{I}_{E,k}^{(N_{R},2)}&\cdots&\mathbf{I}_{E,k}^{(N_{R},N_{ R})}\end{bmatrix} \in\mathbb{H}_{+}^{N_{T}N_{R}},\\ \mathbf{J}_{E,k}^{(k,k)}\in\mathbb{H}_{+}^{N_{R}},\mathbf{K}_{E,k}^{(k,k)}\in\mathbb{H}_{+ }^{N_{R}}.\end{split} \tag{29}\]  
Then, by utilizing (27)-(29), the Lagrange dual function can be rewritten as \[