# Energy-Efficient Power Allocation in Non-Linear Energy Harvesting Multiple Relay Systems
## 3 Joint Optimization Algorithm for the Relay Transmit Power and PS Ratio  
The optimization problem (P1) is a multi-variable joint optimization problem and a complex NP problem. Because of the piecewise linear EH model, the optimization problem has different results for its different polyline segments. In order to solve (P1), there are two main steps as follows. In the first step, the power allocation problem of multiple relays is decoupled into a single-relay power allocation problem by using the fixed-point iteration algorithm, which assumes that only the power of this relay is to be allocated and that the power allocation of other relays is known. For convenience, some iteration variables \(\psi_{\mathrm{i}}=\sum\limits_{w=1,w\
eq i}^{N}\gamma_{w}\), \(\varphi_{i}=\sum\limits_{w=1,w\
eq i}^{N}P_{w}\) and \(\phi_{i}=\sum\limits_{w=1,w\
eq i}^{N}\big{(}P_{w}+P_{Q}-P_{H}^{op}(\rho_{w}) \big{)}\) are introduced. In the second step, we assume \(\rho_{i}=1\) and calculate the maximum number of segments which \(P_{RF}^{i}(\rho_{i})\) may belong to, expressed as \(S_{i}(s_{i}\in\{0,1,\ldots,S_{i}\},i=1,2,\ldots,N)\). In a given number of segments \(s_{i}\), we transform this non-convex problem into a convex problem by subtraction and find a closed-form solution. According to the Dinkelbach method [19], the maximum EE \(x^{*}=\underset{P_{i}\rho_{i}}{\text{max}}\eta=\frac{\frac{1}{2}\log_{2}\big{(} 1+\psi_{\mathrm{i}}+\gamma_{i}\big{(}P_{i}^{*}+\rho_{i}^{*}\big{)}\big{)}}{P_ {S}+2P_{\mathrm{C}}+\phi_{\mathrm{i}}+P_{\mathrm{i}}^{*}+P_{Q}-P_{H}^{i}(\rho_ {i}^{*})}\) is achieved if and only if the following equation is satisfied:  
\[\underset{P_{i}\rho_{i}}{\text{max}}\frac{1}{2}\log_{2}(1+\psi_{\mathrm{i}}+ \gamma_{i}\big{(}P_{i},\rho_{i}\big{)})-x^{*}\cdot\Big{(}P_{\mathrm{S}}+2P_{ \mathrm{C}}+\phi_{i}+P_{i}+P_{Q}-P_{H}^{i}(\rho_{i})\Big{)}=0 \tag{9}\]  
Therefore, we convert the fractional objective function to subtraction for a given segment \(s_{i}\), and (P1) can be rewritten as  
\[ \begin{array}{rl}(\mathrm{P2})&\underset{P_{i}\rho_{i}}{\text{max}}\frac{1 }{2}\log_{2}\big{(}1+\psi_{\mathrm{i}}+\gamma_{i}\big{(}P_{i},\rho_{i}\big{)} \big{)}-x^{*}\cdot\big{(}P_{\mathrm{S}}+2P_{\mathrm{C}}+\phi_{i}+P_{i}+P_{Q}- P_{H}^{i}\big{(}\rho_{i}\big{)}\big{)}\\ s.t.C1-C5\end{array}  \tag{10}\]  
However, (P2) is still a non-convex problem. In order to convert this problem into a convex one, we let \(t_{i}=1-\rho_{i}\) and introduce an auxiliary variable \(r_{i}\), expressed as  
\[r_{i}=\frac{1}{2}\log_{2}\Bigg{(}1+\sum\limits_{i=1}^{N}\gamma_{i}\Bigg{)}= \min\bigg{(}\frac{1}{2}\log_{2}(1+\psi_{\mathrm{i}}+\gamma_{r}^{i}),\frac{1}{ 2}\log_{2}(1+\psi_{\mathrm{i}}+\gamma_{d}^{i})\bigg{)} \tag{11}\]The optimization problem (P2) can be rewritten as  
\[(\mathrm{P3}) \underset{P_{i},t_{i}}{\max} f(P_{i_{r}},t_{i_{r}},r_{i})=r_{i}-x^{*}\cdot\big{(}P_{S}+2P_{C}+\phi_{i}+P_{i}+P_ {i}+P_{Q}-a_{s_{i}}\cdot(1-t_{i})P_{S}h_{i}-b_{s_{i}}\big{)} \tag{12}\] \[s.t.\mathrm{C1}: t_{\min}\leq t_{i}\leq t_{\max}\] \[\mathrm{C2}: 0\leq P_{i}\leq a_{s_{i}}\cdot(1-t_{i})P_{S}h_{i}+b_{s_{i}}-P_{Q}\] \[\mathrm{C3}: r_{i}\geq\frac{1}{2}\log_{2}(1+\gamma_{\min})\] \[\mathrm{C4}: \frac{1}{2}\log_{2}\Bigg{(}1+\psi_{\mathrm{i}}+\frac{t_{i}\cdot P _{S}h_{i}}{t_{i}\cdot\sigma_{a_{i}}^{2}+\sigma_{ID_{i}}^{2}}\Bigg{)}\geq r_{i}\] \[\mathrm{C5}: \frac{1}{2}\log_{2}\Bigg{(}1+\psi_{\mathrm{i}}+\frac{P_{i}S_{i} }{\sigma_{D}^{2}}\Bigg{)}\geq r_{i}\] \[\mathrm{C6}: P_{i}\leq P_{\max}-\varphi_{i}\]  
where \(t_{\min}=1-\frac{P_{i}^{s_{i}+1}}{P_{S}h_{i}}\) and \(t_{\max}=1-\frac{P_{i}^{s_{i}}}{P_{S}h_{i}}\). **Theorem 1**. : The optimization problem (P3) is a convex programming problem. Proof of Theorem 1.: The objective function of (P3) regarding \(P_{i_{r}}\), \(t_{i}\) and \(r_{i}\) is a linear function, and constraints C1, C2, C3 and C6 are linear constraints, while constraints C4 and C5 are nonlinear constraints, so (P2) is a nonlinear programming problem.