# Cooperative Jamming with AF Relay in Power Monitoring and Communication Systems for Mining
## 3 Optimization Problem Transformation
\[-\log\lvert\mathbf{E}+\mathbf{\Gamma}_{1}\rvert=\log\lvert \mathbf{Q}_{E1}\rvert-\log\Bigl{\lvert}(\mathbf{H}_{SE,1}\mathbf{L})^{K}+ \mathbf{Q}_{E1}\Bigr{\rvert}\] \[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]0\[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]1 \[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]2 \[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]3  
Then, the auxiliary variables \(C_{E11}\), \(C_{E12}\), \(C_{E21}\) and \(C_{E22}\) can be rewritten according to Lemma 1 as  
\[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]4  
\[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]5  
\[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]6  
\[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]7  
where  
\[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]8  
\[\text{s.t.}\ \|\mathbf{L}\|^{2}\leqslant P_{S},\ \|\mathbf{J}_{1}\|^{2} \leqslant P_{S},\ \|\mathbf{J}_{2}\|^{2}\leqslant P_{R},\]9  
\[tr((\mathbf{G}\mathbf{H}_{SR,1}\mathbf{L})^{K}+(\mathbf{G} \mathbf{H}_{I}\mathbf{J}_{1})^{K}+\sigma_{R}^{2}\mathbf{G}^{K})\leqslant P_{R} \quad\forall\mathbf{H}_{ij,k}\in\mathcal{H}_{ij,k} \tag{16c}\]0  
\[tr((\mathbf{G}\mathbf{H}_{SR,1}\mathbf{L})^{K}+(\mathbf{G} \mathbf{H}_{I}\mathbf{J}_{1})^{K}+\sigma_{R}^{2}\mathbf{G}^{K})\leqslant P_{R} \quad\forall\mathbf{H}_{ij,k}\in\mathcal{H}_{ij,k} \tag{16c}\]1  
\[tr((\mathbf{G}\mathbf{H}_{SR,1}\mathbf{L})^{K}+(\mathbf{G} \mathbf{H}_{I}\mathbf{J}_{1})^{K}+\sigma_{R}^{2}\mathbf{G}^{K})\leqslant P_{R} \quad\forall\mathbf{H}_{ij,k}\in\mathcal{H}_{ij,k} \tag{16c}\]2  
and note the decomposition \(\mathbf{T}_{E2}=\left[ \begin{array}{cc}\mathbf{T}_{E21}&\mathbf{T}_{E22}& \mathbf{T}_{E23}\end{array} \right]\) and \(\mathbf{X}=\left[ \begin{array}{cc}1&\mathbf{0}\end{array} \right]\in\mathbb{C }^{1\times Nr}\). After substituting (24)-(27) into (16a), the secrecy rate of the system is equivalently rewritten as  
\[\max_{\mathbf{L},\mathbf{J}_{1},\mathbf{J}_{2},\mathbf{G},\mathbf{K}_{i} \succ 0,\mathbf{T}_{i}}\min_{\mathbf{H}_{ijk}\in\mathcal{H}_{ijk}}f(\mathbf{L}, \mathbf{J}_{1},\mathbf{J}_{2},\mathbf{G},\mathbf{S}_{i},\mathbf{D}_{i}) \tag{28}\]  
\[\text{s.t. (\ref{eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq: eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:  
\[\mathbf{\Omega}_{DSR,1}=\left[ \begin{array}{c}\mathbf{L}^{T}\otimes\mathbf{F}_{D} \mathbf{T}_{D}\mathbf{\bar{H}}_{RD,2}\mathbf{G}\\ \mathbf{0}\\ \mathbf{0}\end{array} \right] \tag{39}\]  
\[\mathbf{\Omega}_{DSD,2}=\left[ \begin{array}{c}\left(\mathbf{G}\mathbf{\bar{H}}_ {SR,1}\mathbf{L}\right)^{T}\otimes\mathbf{F}_{D}\mathbf{T}_{D}\\ \left(\mathbf{G}\mathbf{\bar{H}}_{RR,1}\mathbf{J}_{1}\right)^{T}\otimes \mathbf{F}_{D}\mathbf{T}_{D}\\ \sigma_{R}\mathbf{G}^{T}\otimes\mathbf{F}_{D}\mathbf{T}_{D}\\ \mathbf{0}\end{array} \right] \tag{40}\]  
\[\mathbf{\Omega}_{DR,1}=\left[ \begin{array}{c}\mathbf{0}\\ \mathbf{J}_{1}^{T}\otimes\mathbf{F}_{D}\mathbf{D}_{D}\mathbf{\bar{H}}_{RD,2} \mathbf{G}\\ \mathbf{0}\end{array} \right] \tag{41}\]  
Then, we exploit the Schur complement lemma to recast the constraint (31) as a matrix inequality by substituting (35) and (37). \[ \begin{bmatrix}\beta_{D}&\bar{\phi}_{D}^{H}\\ \bar{\phi}_{D}&\mathbf{E}\end{bmatrix} \succ-