## 3 Scheme Design
### Iterative Algorithm Based on FPP-SCA
#### 3.1.1 Optimization of Relay Beamformer \(\mathbf{F}_{r}\)  
Then, the optimization of \(\mathbf{F}_{r}\) with a fixed \(\mathbf{F}_{i}\) and \(\mathbf{W}_{i}\) is discussed. According to [42] (p. 77),  
\[\mathit{Tr}(\mathbf{ABCD})=\left(vec(\mathbf{D}^{T})\right)^{T}(\mathbf{C}^{T }\otimes\mathbf{A})vec(\mathbf{B}), \tag{13}\]  
where \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), and \(\mathbf{D}\) are arbitrary matrices with compatible dimensions, \(\otimes\) is the Kronecker product, and \(vec(\cdot)\) represents the matrix vectorization operator. To guarantee the feasibility of Equation (11), the feasible region is relaxed and approximated by adding slacks \(\mathbf{s}\in\mathbb{R}^{2}\) to the non-convex constraint of Equation (11d), and the positive slack variables and slack penalty are used in Equation (11a). Then, the original problem can be recast as  
\[\min_{\mathbf{f}_{r}} \mathbf{f}_{r}^{H}\mathbf{Z}_{r}\mathbf{f}_{r}-\mathbf{Q}_{r}^{H }\mathbf{f}_{r}-\mathbf{f}_{r}^{H}\mathbf{Q}_{r}+\mathbf{C}_{r}+\lambda\left\| \mathbf{s}\right\| \tag{14a}\] \[\mathrm{s.t.} \mathbf{f}_{r}^{H}\mathbf{Q}_{r}\mathbf{f}_{\varsigma}\leqslant- \xi_{i}+s_{m}\] (14b) \[\mathbf{f}_{r}^{H}\mathbf{Q}\mathbf{f}_{\varsigma}\leqslant p_{r}\] (14c) \[s_{m}\geqslant 0,\;\;m=1,2, \tag{14d}\]  
where \(\left\|\cdot\right\|\) can be any vector norm, \(\left\|\mathbf{s}\right\|\) denotes the slack penalty term, and \(\lambda\geqslant 1\) is the trade-off between the original objective function and \(\left\|\mathbf{s}\right\|\). Besides, \(\mathbf{f}_{r}=vec(\mathbf{F}_{r})\), \(\mathbf{Z}_{r}=\mathbf{Z}_{r1}+\mathbf{Z}_{r2}\), \(\mathbf{Q}_{r}=\mathbf{Q}_{r1}+\mathbf{Q}_{r2}\), \(\mathbf{C}_{r}=\mathbf{C}_{r1}+\mathbf{C}_{r2}\), \(\mathbf{Z}_{ri}=\beta_{i}\left(\mathbf{J}_{i_{C}}\mathbf{J}_{i_{C}}^{H}+\sigma_ {r}^{2}\mathbf{I}_{N_{R}}\right)^{T}\otimes\left(\mathbf{H}_{RS_{i}}^{H} \mathbf{W}_{i}^{H}\mathbf{W}_{i}\mathbf{H}_{RS_{i}}\right)\), \(\mathbf{Q}_{ri}=\sqrt{\beta_{i}}vec\left(\mathbf{J}_{i_{C}}^{H}\mathbf{H}_{RS _{i}}^{H}\mathbf{W}_{i}^{H}\right)\), \(\mathbf{C}_{ri}=\mathit{Tr}\left(\beta_{i}\sigma_{S_{i}}^{2}\mathbf{W}_{i} \mathbf{W}_{i}^{H}+\mathbf{I}_{M_{R}}\right)\), \(\mathbf{Q}=\mathbf{I}_{N_{R}}\otimes\mathbf{R}^{T}\), \(\mathbf{Q}_{ir}=-\mathbf{H}_{RS_{i}}^{H}\mathbf{H}_{RS_{i}}\otimes\mathbf{R}^ {T}\), and \(\xi_{i}=\frac{\pi_{i}}{(1-\beta_{i})}-\mathit{Tr}\left(\sigma_{S_{i}}^{2} \mathbf{I}_{M_{R}}\right)\). Since \(\mathbf{Q}_{ir}\) is negative, Equation (14) is non-convex. To tackle this subproblem, we define \(g(\mathbf{f}_{r})=\mathbf{f}_{r}^{H}\mathbf{Q}_{ir}\mathbf{f}_{r}\) and assume that a center point \(\mathbf{z}_{r}\in\mathbb{C}^{N\times 1}\), \(N=N_{T}\times N_{R}\) is given. Introducing \(\tilde{g}(\mathbf{f}_{r})\triangleq 2 Re\left(\mathbf{z}_{r}^{H}\mathbf{Q}_{ir} \mathbf{f}_{r}\right)-\mathbf{z}_{r}^{H}\mathbf{Q}_{ir}\mathbf{z}_{r}\), Theorem 1 can be established and proved. **Theorem 1**. : \(\tilde{g}(\mathbf{z}_{r})=g(\mathbf{z}_{r})\)_; (ii) \(\tilde{g}(\mathbf{f}_{r})\geq g(\mathbf{f}_{r})\); and (iii) \(\partial\tilde{g}(\mathbf{f}_{r})/\partial\mathbf{f}_{r}|_{\mathbf{f}_{r}= \mathbf{z}_{r}}=\partial g(\mathbf{f}_{r})/\partial\mathbf{f}_{r}|_{\mathbf{f}_{ r}=\mathbf{z}_{r}}\)._  
Proof. : Substituting \(\mathbf{z}_{r}\) into \(\tilde{g}(\mathbf{f}_{r})\) and \(g(\mathbf{f}_{r})\), (i) can be easily certified. For (ii), \(\mathbf{Q}_{ir}\preceq 0\), \((\mathbf{f}_{r}-\mathbf{z}_{r})^{H}\mathbf{Q}_{ir}(\mathbf{f}_{r}-\mathbf{z}_{ r})\leqslant 0\) always holds, which shows\(2 Re\left(\mathbf{z}_{r}^{H}\mathbf{Q}_{ir}\mathbf{f}_{r}\right)-\mathbf{z}_{r}^{H} \mathbf{Q}_{ir}\mathbf{z}_{r}=\tilde{g}(\mathbf{f}_{r})\), property (ii) is proved. For (iii), the derivatives can be computed as \(\partial\tilde{g}(\mathbf{f}_{r})/\partial\mathbf{f}_{r}|_{\mathbf{f}_{r}= \mathbf{z}_{r}}=(\mathbf{z}_{r}^{H}\mathbf{Q}_{ir})^{T}\) and \(\partial g(\mathbf{f}_{r})/\partial\mathbf{f}_{r}|_{\mathbf{f}_{r}=\mathbf{z }_{r}}=(\mathbf{z}_{r}^{H}\mathbf{Q}_{ir})^{T}\); therefore, property (iii) is proved.