## 3 Solution of the Optimization Problem
### Semi-Definite Programming Relaxation
\begin{bmatrix}\mathbf{A}_{2}&\mathbf{b}_{2}\\ \mathbf{b}_{2}^{H}&c_{2}\end{bmatrix} \succeq\mathbf{0} \tag{16}\]  
_provided that there exsists a vector \(\mathbf{\hat{x}}\) such that we obtain \(f_{i}(\mathbf{\hat{x}})<0\)._  
As a result, we can perform Lemma 1 to constraint (C2). Particularly, we substitute \(\mathbf{G}_{k}=\bar{\mathbf{G}}_{k}+\Delta\mathbf{G}_{k}\) in constraint C2. Hence, the implication  
\[ \begin{split}&\Delta g_{k}^{H}\Delta g_{k}\leq v_{k}^{2}\Rightarrow,\\ &\Delta g_{k}^{H}\Delta_{2}\Delta g_{k}+2 Re\left\{\bar{g}_{k}^{H} \Delta_{2}\Delta g_{k}\right\}+\bar{g}_{k}^{H}\Delta_{2}\bar{g}_{k}-r_{k} \sigma_{E,k}^{2}\leq 0\end{split}  \tag{17}\]  
holds if and only if there exist \(\bar{\lambda}_{k}\geq 0\), \(k\in\{1,2,\cdots,K_{E}\}\), such that the following linear matrix inequalities hold:  
\[ \begin{split}\text{C2}:& N_{E,k}\big{(}\mathbf{W},\mathbf{V} _{0},\bar{\lambda}_{k}\big{)}\\ &=\begin{bmatrix}\bar{\lambda}_{k}\mathbf{I}_{N_{T}N_{R}}-\mathbf{\Delta} _{2}&-\mathbf{\Delta}_{2}\bar{g}_{k}\\ \bar{g}_{k}^{H}\mathbf{\Delta}_{2}&-\bar{g}_{k}^{H}\mathbf{\Delta}_{2}\bar{g}_{k}-v_ {k}^{2}\bar{\lambda}_{k}+r_{k}\sigma_{E,k}^{2}\end{bmatrix}\succeq\mathbf{0},\end{split}  \tag{18}\]  
where \(\bar{\mathbf{W}}=\mathbf{I}_{N_{R}}\otimes\mathbf{W}\), \(\bar{\mathbf{V}}_{0}=\mathbf{I}_{N_{R}}\otimes\mathbf{V}_{0}\), \(\Delta g_{k}=vec(\mathbf{\Delta G}_{k})\), \(\bar{g}_{k}=vec(\bar{\mathbf{G}}_{k})\) and \(\mathbf{\Delta}_{2}=\bar{\mathbf{W}}-r_{k}\bar{\mathbf{V}}_{0}\). Similarly, by applying Lemma 1, we can equivalently write the constraint (C9) as  
\[ \begin{split}\text{C9}:& T_{E,k}\big{(}\mathbf{W},\mathbf{V} _{0},\mathbf{V},\bar{\mu}_{k},t\big{)}\\ &=\begin{bmatrix}\bar{\mu}_{k}\mathbf{I}_{N_{T}N_{R}}+\mathbf{\Delta}_{3}& \mathbf{\Delta}_{3}\bar{g}_{k}\\ \bar{g}_{k}^{H}\mathbf{\Delta}_{3}&\bar{g}_{k}^{H}\mathbf{\Delta}_{3}\bar{g}_{k}-v_ {k}^{2}\bar{\mu}_{k}-\tau_{E,k}\end{bmatrix}\succeq\mathbf{0}\end{split}  \tag{19}\]  
for \(\bar{\mu}_{k}\geq 0\), \(k\in\{1,2,\cdots,K_{E}\}\), where \(\bar{\mathbf{V}}=\mathbf{I}_{N_{R}}\otimes\mathbf{V}\) and \(\mathbf{\Delta}_{3}=\bar{\mathbf{W}}+\bar{\mathbf{V}}+\bar{\mathbf{V}}_{0}\). Therefore, substituting (18) and (19) back into (15)  
\[ \begin{split}&\max_{\mathbf{W},\mathbf{V}_{0},\rho,t,\tau_{E,k}\bar{ \lambda},\bar{\mathbf{\rho}}}t\\ s.t.&\text{C1, C3, C4, C5, C6, C7,}\\ &\text{C2}:N_{E,k}\big{(}\mathbf{W},\mathbf{V}_{0},\bar{\lambda}_{k} \big{)}\succeq\mathbf{0},\forall k,\\ &\text{C8}:\ \text{rank}(\mathbf{W})=1,\\ &\text{C9}:T_{E,k}\big{(}\mathbf{W},\mathbf{V}_{0},\mathbf{V},\bar{\mu}_{k},t \big{)}\succeq\mathbf{0},\forall k,\\ &\text{C10}:\ \bar{\lambda}_{k}\geq 0,\ \bar{\mu}_{k}\geq 0,\ \forall k, \end{split}  \tag{20}\]  
where \(\bar{\mathbf{\lambda}}\) and \(\bar{\mathbf{\rho}}\) denote auxiliary optimization variable vectors, whose elements \(\bar{\lambda}_{k}\), \(k\in\{1,2,\cdots,K_{E}\}\), and \(\bar{\mu}_{k}\), \(k\in\{1,2,\cdots,K_{E}\}\) were introduced in (18) and (19), respectively. Now, (C8): \(\text{rank}(\mathbf{W})=1\) is the main obstacle in solving the optimization problem (20). By relaxing constraint (C8), i.e., dropping the rank constraint from the optimization problem (20), we can reexpress the optimization problem (20) as follows:  
\[ \begin{split}&\max_{\mathbf{W},\mathbf{V}_{0},\rho,t,\tau_{E,k}\bar{ \lambda},\bar{\mathbf{\rho}}}t\\ s.t.&\ \ \ \ \ \ \text{C1, C2, C3, C4, C5, C6, C7, C9, C10.}\end{split}  \tag{21}\]It is worth noting that the relaxed optimization problem (21) becomes a convex SDP problem that can be solved efficiently by a numerical solver such as \(SDPT3\) and \(SeDuMi\). Based on the basic principles of convex optimization theory, if the obtained solution \(\mathbf{W}\) for optimization problem (21) admits a rank-one matrix, then \(\mathbf{W}\) is the optimal solution of the optimization problem (20). Then, we can obtain the optimal information vector \(\mathbf{w}\) by performing eigenvalue decomposition on \(\mathbf{W}\).