## 3 Solution of the Optimization Problem
### Optimality Conditions for SDP Relaxation
\begin{split}&\mathcal{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}}\operatorname{Tr}\bigl{[}\mathbf{X}_{k}\bar{ \mathbf{\Lambda}}_{E,k}+\sum_{n=1}^{N_{R}}(r_{k}\mathbf{V}_{0}-\mathbf{W})\mathbf{K}_{E,k}^{(n,n)}\bigr{]}\\ &+\sum_{k=1}^{K_{E}}\operatorname{Tr}\bigl{[}\mathbf{Y}_{k}\mathbf{ \Lambda}_{E,k}+\sum_{n=1}^{N_{R}}\big{(}\mathbf{W}+\mathbf{V}_{0}\big{)}\mathbf{f}_{E,k}^{ (n,n)}\bigr{]}\\ &+\alpha\bigl{[}P-\operatorname{Tr}\bigl{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_ {0}\bigr{)}\bigr{]}+\beta\bigl{[}\operatorname{Tr}\bigl{(}\big{(}\mathbf{W}-r\mathbf{V }-r\mathbf{V}_{0}\big{)}\mathbf{H}\bigr{)}-\frac{\sigma_{I}^{2}r}{\rho}\bigr{]}\\ &+\gamma\bigl{[}\operatorname{Tr}\bigl{(}\big{(}\mathbf{W}+\mathbf{V}+ \mathbf{V}_{0}\big{)}\mathbf{H}\bigr{)}-\frac{\omega_{I}}{1-\rho}\bigr{]}+ \operatorname{Tr}\bigl{(}\mathbf{\Theta}\mathbf{W}\bigr{)}+\operatorname{Tr}\bigl{(} \mathbf{\Xi}\mathbf{V}\bigr{)}+\operatorname{Tr}\bigl{(}\mathbf{\Omega}\mathbf{V}_{0}\bigr{)} \\ &+\sum_{k=1}^{K_{E}}\zeta_{k}\bigl{[}\frac{M_{E,k}}{1+\exp\bigl{(} -a_{E,k}\bigl{(}\tau_{E,k}-b_{E,k}\bigr{)}\bigr{)}}-X_{E,k}\bigl{(}Y_{E,k}+t \bigr{)}\bigr{]}.\end{split}  \tag{30}\]  
It can be verified that the relax optimization problem (20) is jointly convex with regard to the primal variables and satisfies the [PERSON]'s constraint qualification. Therefore, there is no gap between the primal optimization problem and dual optimization problem. That is to say, strong duality holds and solving the dual optimization problem is equivalent to solving the primal problem. Considering (30), the dual optimization problem is expressed as:  
\[\min_{\{\mathbf{X}_{k}\succeq\mathbf{0}\},\{\mathbf{Y}_{k}\succeq\mathbf{0}\},\mathbf{\Theta}, \mathbf{\Xi},\mathbf{\Omega}\succeq\mathbf{0},\mathbf{\beta},\gamma\succeq\mathbf{0},\{\bar{\zeta }_{k}\succeq\mathbf{0}\},\mathbf{\Pi}, \tag{31}\]  
where \(\mathbf{\Pi}\triangleq\bigl{\{}\mathbf{W},\mathbf{V},\mathbf{V}_{0},\mathbf{X}_{k},\mathbf{Y}_{k},\alpha,\beta,\gamma,\zeta_{k},\mathbf{\Theta},\mathbf{\Xi},\mathbf{\Omega},\rho,t,\tau_{E,k},\bar {\lambda}_{k},\bar{\mu}_{k}\bigr{\}}\). From (31), we have \(\bigl{\{}\mathbf{X}_{k}^{*},\mathbf{Y}_{k}^{*},\mathbf{\Theta}^{*},\mathbf{\Xi}^{*},\mathbf{\Omega} ^{*},\alpha^{*},\beta^{*},\gamma^{*},\zeta_{k}^{*}\bigr{\}}\) and \(\bigl{\{}\mathbf{W}^{*},\mathbf{V}_{0}^{*},\rho^{*},\tau_{E,k}^{*},t^{*}\bigr{\}}\) as the dual and the primal optimal solution of (20) and (31), respectively. Considering (31), the Karush-Kuhn-Tucker (KKT) conditions for the optimal \(\mathbf{W}^{*}\) are given by  
\[\mathbf{\Theta}^{*}\succeq\mathbf{0},\alpha\geq 0,\beta\geq 0,\gamma\geq 0, \tag{32}\]  
\[\mathbf{\Theta}^{*}\mathbf{W}^{*}=\mathbf{0}, \tag{33}\]  
\[\bigl{[}\sum_{k=1}^{K_{E}}\sum_{n=1}^{N_{R}}\big{(}\mathbf{I}_{E,k}^{(n,n)(*)}- \mathbf{K}_{E,k}^{(n,n)(*)}\bigr{)}+\bigl{(}\beta^{*}+\gamma^{*}\bigr{)}\mathbf{H}- \alpha^{*}\mathbf{I}_{N_{T}}\bigr{]}\mathbf{W}^{*}+\mathbf{\Theta}^{*}=\mathbf{0}, \tag{34}\]  
\[\frac{\partial\mathcal{L}(\mathbf{\Pi})}{\partial\rho^{*}}=0\Rightarrow\ \rho^{*}= \frac{\sqrt{\beta^{*}r\sigma_{I}^{2}}}{\sqrt{\gamma^{*}\omega_{I}+\sqrt{\beta^{ *}r\sigma_{I}^{2}}}}. \tag{35}\]  
For notational simplicity, we define  
\[\mathbf{R}^{*}=\sum_{k=1}^{K_{E}}\sum_{n=1}^{N_{R}}\big{(}\mathbf{I}_{E,k}^{(n,n)(*)}- \mathbf{K}_{E,k}^{(n,n)(*)}\bigr{)}+\bigl{(}\beta^{*}+\gamma^{*}\bigr{)}\mathbf{H}- \alpha^{*}\mathbf{I}_{N_{T}}. \tag{36}\]  
Then, we have  
\[\mathbf{R}^{*}=\mathbf{T}^{*}+\bigl{(}\beta^{*}+\gamma^{*}\bigr{)}\mathbf{H}. \tag{37}\]  
However, in general, we define \(\bar{r}=\operatorname{rank}(\mathbf{T}^{*})\). Then, we consider two cases of \(\bar{r}\) with the aim of analyzing \(\mathbf{R}^{*}\). Firstly, it is assumed that \(\bar{r}=N_{T}\), i.e., the matrix \(\mathbf{T}^{*}\) is full rank. Upon that, we obtain  
\[