## 3 Solution of the Optimization Problem
### Semi-Definite Programming Relaxation  
In order to solve the optimization problem (13), we introduce a slack variable \(t\geq 0\). With the slack variable \(t\), the problem (13) can be equivalently formulated as the following optimization problem:  
\[ \begin{split}\max_{\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho,t}& t\\ s.t.&\text{C1}:\operatorname{Tr}\bigl{[}\bigl{(} \mathbf{W}-r\mathbf{V}_{0}-r\mathbf{V}\bigr{)}\mathbf{H}\bigr{]}\geq\frac{\sigma_{I}^{2}r}{\rho},\\ &\text{C2}:\max_{\mathbf{\alpha}_{\mathbf{G}_{k}}\in\mathbb{E}_{k}} \operatorname{Tr}\bigl{[}\mathbf{G}_{k}^{H}\bigl{(}\mathbf{W}-r_{k}\mathbf{V}_{0}\bigr{)} \mathbf{G}_{k}\bigr{]}\leq r_{k}\sigma_{E,k}^{2},\forall k,\\ &\text{C3}:\Phi_{I}\bigl{(}\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho\bigr{)} \geq E_{\min},\\ &\text{C4}:\operatorname{Tr}\bigl{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_{0} \bigr{)}\leq P,\\ &\text{C5}:\min_{\mathbf{\alpha}_{\mathbf{G}_{k}}\in\mathbb{E}_{k}} \operatorname{E}_{E,k}^{\text{Nonlinear}}\geq t,\forall k,\\ &\text{C6}:0\leq\rho\leq 1,\quad\quad\text{C7}:\mathbf{W},\mathbf{V},\mathbf{V }_{0}\succeq\mathbf{0},\\ &\text{C8}:\text{Rank}\bigl{(}\mathbf{W}\bigr{)}=1.\end{split}  \tag{14}\]  
To facilitate the optimization problem (14), another slack variable \(\tau_{k}\), \(\forall k\in\{1,\cdots,K_{E}\}\) is also introduced, we can rewrite the problem (14) as  
\[ \begin{split}\max_{\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho,t,\mathbf{\tau}}& t\\ s.t.&\text{C1}:\operatorname{Tr}\bigl{[}\bigl{(} \mathbf{W}-r\mathbf{V}_{0}-r\mathbf{V}\bigr{)}\mathbf{H}\bigr{]}\geq\frac{\sigma_{I}^{2}r}{ \rho},\\ &\text{C2}:\max_{\mathbf{\alpha}_{\mathbf{G}_{k}}\in\mathbb{E}_{k}} \operatorname{Tr}\bigl{[}\mathbf{G}_{k}^{H}\bigl{(}\mathbf{W}-r_{k}\mathbf{V}_{0}\bigr{)} \mathbf{G}_{k}\bigr{]}\leq r_{k}\sigma_{E,k}^{2},\forall k,\\ &\text{C3}:\operatorname{Tr}\bigl{[}\bigl{(}\mathbf{W}+\mathbf{V}+\mathbf{V} _{0}\bigr{)}\mathbf{H}\bigr{]}\geq\frac{\omega_{I}}{1-\rho},\\ &\text{C4}:\operatorname{Tr}\bigl{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_{0} \bigr{)}\leq P,\\ &\text{C5}:\frac{M_{E,k}}{1+\exp\bigl{(}-\alpha_{E,k}\bigl{(} \tau_{E,k}-b_{E,k}\bigr{)}\bigr{)}}\geq X_{E,k}\bigl{(}Y_{E,k}+t\bigr{)}, \forall k,\\ &\text{C6}:0\leq\rho\leq 1,\quad\text{C7}:\mathbf{W},\mathbf{V},\mathbf{V }_{0}\succeq\mathbf{0},\\ &\text{C8}:\text{Rank}\bigl{(}\mathbf{W}\bigr{)}=1,\\ &\text{C9}:\min_{\mathbf{\alpha}_{\mathbf{G}_{k}}\in\mathbb{E}_{k}} \operatorname{Tr}\bigl{(}\mathbf{G}_{k}^{H}\bigl{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_{0}\bigr{)} \mathbf{G}_{k}\bigr{)}\geq\tau_{E,k},\end{split}  \tag{15}\]  
where \(\omega_{I}=b_{I}-\frac{\ln\left\{\frac{M_{I}}{\bigl{(}\varepsilon_{\min}+V_{I }\bigr{)}x_{I}}-1\right\}}{a_{I}}\); \(\mathbf{W}\succeq\mathbf{0}\), \(\mathbf{W}\in\mathbb{H}_{+}^{N_{T}}\) and \(\text{rank}(\mathbf{W})=1\) are imposed to guarantee that \(\mathbf{W}=\mathbf{w}\mathbf{w}^{H}\) holds after optimizing \(\mathbf{W}\). After some manipulations, it can be easily known that constraints (C1) and (C3) are jointly convex with regard to \(\{\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho\}\). Then, we deal with constraints (C2), (C5) and (C9). Although constraints (C1) and (C3) are jointly convex with regard to the optimization variables, they are semi-infinite constraints that are generally intractable for beamforming scheme design. To facilitate this, we can transform them into linear matrix inequalities using the following lemma. Lemma 1: \((S-\text{Procedure})\) _Let \(f_{m}(\mathbf{x})=\mathbf{x}^{H}\mathbf{A}_{m}\mathbf{x}+2 Re\left\{\mathbf{b}_{m}^{H}\mathbf{x} \right\}+c_{m}\), \(m\in\{1,2\}\), where \(\mathbf{x}\in\mathbb{C}^{N\times 1}\), \(\mathbf{A}_{m}\in\mathbb{H}^{N\times N}\), \(\mathbf{b}_{m}\in\mathbb{C}^{N\times 1}\) and \(c_{m}\in\mathbb{R}\). Then, the expression \(f_{1}(\mathbf{x})\leq 0\Rightarrow f_{2}(\mathbf{x})\) holds if and only if there exists a \(\lambda\geq 0\) such that we get:_\[\lambda \begin{bmatrix}\mathbf{A}_{1}&\mathbf{b}_{1}\\ \mathbf{b}_{1}^{H}&c_{1}\end{bmatrix} -