# Energy-Efficient Power Allocation in Non-Linear Energy Harvesting Multiple Relay Systems
## 3 Joint Optimization Algorithm for the Relay Transmit Power and PS Ratio
Because linear functions are both convex and concave, the objective function \(f(P_{i},t_{i},r_{i})\) is convex, and constraints C1, C2, C3 and C6 are concave constraints. By letting \(f_{1}(t_{i})=\frac{1}{2}\log_{2}\Big{(}1+\psi_{\mathrm{i}}+\frac{t_{i}\cdot P _{S}h_{i}}{t_{i}\cdot\sigma_{a_{i}}^{2}+\sigma_{ID_{i}}^{2}}\Big{)}\) and \(f_{2}(P_{i})=\frac{1}{2}\log_{2}\bigg{(}1+\psi_{\mathrm{i}}+\frac{P_{S}h_{i}} {\sigma_{D}^{2}}\bigg{)}\), then we have \(\frac{\partial^{2}f_{1}(t_{i})}{\partial t_{i}^{2}}=\frac{-P_{S}h_{i}\sigma_{ ID_{i}}^{2}\cdot(2a_{t_{i}}t_{i}+b_{t})}{\big{(}a_{t}\cdot t_{i}^{2}+b_{t}\cdot t _{i}+c_{t}\big{)}^{2}\cdot 2\ln(2)}<0\), where \(a_{t}=\Big{(}P_{S}h_{i}+(1+\psi_{\mathrm{i}})\cdot\sigma_{a_{i}}^{2}\Big{)} \cdot\sigma_{a_{t}}^{2}\), \(b_{t}=\Big{(}P_{S}h_{i}+2(1+\psi_{\mathrm{i}})\sigma_{a_{i}}^{2}\Big{)}\), \(\cdot\sigma_{ID_{i}}^{2}\) and \(c_{t}=(1+\psi_{\mathrm{i}})\cdot\sigma_{ID_{i}}^{4}\). Therefore, \(f_{1}(t_{i})\) is a concave function for \(t_{i}\), and \(f_{2}(P_{i})\) is also a concave function for \(P_{i}\) ( i.e., the constraints C4 and C5 are concave constraints). From the definition of convex programming, the optimization problem (P3) is a convex programming problem. \(\square\)  
As Equation (11) is composed of two formulas, (P3) is a maximum-minimum function problem and can be decomposed into two maximization problems to obtain the optimal solution by adding the corresponding constraints, for generality, we assume that all noise variances are equal (i.e., \(\sigma_{a_{i}}^{2}=\sigma_{ID_{i}}^{2}=\sigma_{D}^{2}=\sigma^{2}\)). 1. If \(\gamma_{r}^{i}\geq\gamma_{d}^{i}\) (i.e., \(P_{i}\leq\frac{t_{i}P_{S}h_{i}}{(t_{i}+1)g_{i}}\)), we have \(r_{i}=\frac{1}{2}\log_{2}\big{(}1+\psi_{\mathrm{i}}+\gamma_{d}^{i}\big{)}=\frac{ 1}{2}\log_{2}\Big{(}1+\psi_{\mathrm{i}}+\frac{P_{S}h_{i}}{\sigma^{2}}\Big{)}\). The optimization problem can be expressed as  
\[(\mathrm{P4})\underset{P_{i},t_{i}}{\max} y_{1}(P_{i},t_{i})=\frac{1}{2}\log_{2}\Big{(}1+\psi_{\mathrm{i}}+ \frac{P_{S}h_{i}}{\sigma^{2}}\Big{)} \tag{13}\] \[-x^{*}\cdot\big{(}P_{S}+2P_{C}+\phi_{i}+P_{i}+P_{i}+P_{Q}-a_{s_{i} }\cdot(1-t_{i})P_{S}h_{i}-b_{s_{i}}\big{)}\] \[(\mathrm{P3}) \underset{P_{i},t_{i}}{\max} f(P_{i_{r}},t_{i_{r}},r_{i})=r_{i}-x^{*}\cdot\big{(}P_{S}+2P_{C}+\phi_{i}+P_{i}+P_ {i}+P_{Q}-a_{s_{i}}\cdot(1-t_{i})P_{S}h_{i}-b_{s_{i}}\big{)} \tag{12}\]0 \[(\mathrm{P3}) \underset{P_{i},t_{i}}{\max} f(P_{i_{r}},t_{i_{r}},r_{i})=r_{i}-x^{*}\cdot\big{(}P_{S}+2P_{C}+\phi_{i}+P_{i}+P_ {i}+P_{Q}-a_{s_{i}}\cdot(1-t_{i})P_{S}h_{i}-b_{s_{i}}\big{)} \tag{12}\]1 \[(\mathrm{P3}) \underset{P_{i},t_{i}}{\max} f(P_{i_{r}},t_{i_{r}},r_{i})=r_{i}-x^{*}\cdot\big{(}P_{S}+2P_{C}+\phi_{i}+P_{i}+P_ {i}+P_{Q}-a_{s_{i}}\cdot(1-t_{i})P_{S}h_{i}-b_{s_{i}}\big{)} \tag{12}\]2 \[(\mathrm{P3}) \underset{P_{i},t_{i}}{\max} f(P_{i_{r}},t_{i_{r}},r_{i})=r_{i}-x^{*}\cdot\big{(}P_{S}+2P_{C}+\phi_{i}+P_{i}+P_ {i}+P_{Q}-a_{s_{i}}\cdot(1-t_{i})P_{S}h_{i}-b_{s_{i}}\big{)} \tag{12}\]3 \[(\mathrm{P3}) \underset{P_{i},t_{i}}{\max} f(P_{i_{r}},t_{i_{r}},r_{i})=r_{i}-x^{*}\cdot\big{(}P_{S}+2P_{C}+\phi_{i}+P_{i}+P_ {i}+P_{Q}-a_{s_{i}}\cdot(1-t_{i})P_{S}h_{i}-b_{s_{i}}\big{)} \tag{12}\]42. If \(\gamma_{r}^{i}\leq\gamma_{d}^{i}\) (i.e., \(P_{i}\geq\frac{t_{i}P_{5}h_{i}}{(t_{i}+1)g_{i}}\)), we have \(r_{i}=\frac{1}{2}\log_{2}\bigl{(}1+\psi_{1}+\gamma_{r}^{i}\bigr{)}=\frac{1}{2} \log_{2}\)\(\Bigl{(}1+\psi_{1}+\frac{t_{i}P_{5}h_{i}}{(t_{i}+1)\sigma^{2}}\Bigr{)}\). The optimization problem can be expressed as \[