# Joint Optimization of Relay Communication Rates in Clustered Drones under Interference Conditions
## 5 Joint Optimization Scheme for Communication Rate
### Optimization of Power Allocation Factor for Cluster Drones  
To study the optimization problem of the power allocation factor \(\lambda\) for cluster drones, it is necessary to fix variables such as the recently updated modulation method \(U^{(c)}(k)\) of the relay drone, the transmission power \(P^{R,(c)}(k)\), and the three-dimensional trajectory \(Q^{(c)}(k)\) during the \(c\)th round within time slot \(k\). Given \(\partial SNR_{i,S}/\partial SINR_{i,R}^{Ruary}\geq 0\), and with the other variables held constant, the optimization of the transmission power for cluster drones across different time slots is independent [29]. Let \(\varphi\) be the new objective function value, with its expression as follows:  
\[\varphi^{(c)}=log_{2}(U^{(c)}(k))B_{n}^{Ruary}\sum\limits_{i=1}^{M}log_{2}(1+ \frac{\lambda_{i}^{(c)}(k)P^{F}(k)H_{i}^{w,(c)}(k)}{\sum\limits_{j\in\Gamma} \lambda_{j}^{(c)}(k)P^{F}(k)H_{j}^{w,(c)}(k)+n_{0}B_{n}^{Ruary}}) \tag{66}\]  
However, the objective function \(\varphi^{(c)}\) is still not a convex function and the problem is still difficult to solve; fortunately, Equation (66) is satisfied, as follows:  
\[ \begin{split}\varphi^{(c)}=log_{2}(U^{(c)}(k))B_{n}^{Ruary}\sum \limits_{i=1}^{M}log_{2}(1+\frac{\lambda_{i}^{(c)}(k)P^{F}(k)H_{i}^{w,(c)}(k) }{\sum\limits_{j\in\Gamma}\lambda_{j}^{(c)}(k)P^{F}(k)H_{j}^{w,(c)}(k)+n_{0}B_ {n}^{Ruary}})\\ \lambda_{i}^{(c)}(k)P^{F}(k)H_{i}^{w,(c)}(k)+\sum\limits_{j\in \Gamma}\lambda_{j}^{(c)}(k)P^{F}(k)H_{j}^{w,(c)}(k)\\ =log_{2}(U^{(c)}(k))B_{n}^{Ruary}log_{2}(\frac{\sum\limits_{j\in \Gamma\cup\{i\}}\lambda_{i}^{(c)}(k)P^{F}(k)H_{i}^{w,(c)}(k)+n_{0}B_{n}^{Ruary}} {n_{0}B_{n}^{Ruary}})\\ \times\frac{\sum\limits_{j\in\Gamma}\lambda_{j}^{(c)}(k)P^{F}(k)H _{j}^{w,(c)}(k)+n_{0}B_{n}^{Ruary}}{\sum\limits_{l\in\Gamma/\{j\}}\lambda_{l}^ {(c)}(k)P^{F}(k)H_{l}^{w,(c)}(k)+n_{0}B_{n}^{Ruary}})\\ =log_{2}(U^{(c)}(k))B_{n}^{Ruary}log_{2}(\frac{\sum\limits_{j\in \Gamma\cup\{i\}}\lambda_{i}^{(c)}(k)P^{F}(k)H_{i}^{w,(c)}(k)+n_{0}B_{n}^{Ruary} }{n_{0}B_{n}^{Ruary}})=\widetilde{\varphi}^{(c)}\end{split}  \tag{67}\]  
Thus, the original optimization problem \(C_{1}\) can be varied as follows:  
\[ \begin{split} C_{9}:\max\limits_{\{\lambda\}}:\widetilde{ \varphi}^{(c)}\\ (22),(23),(31),(33),(39)\end{split}  \tag{68}\]  
In the above optimization problem \(C_{9}\), the objective function \(\widetilde{\varphi}^{(c)}\) is a concave function of the optimization variables \(\lambda^{(c)}(k)\). Proof: Take the second order derivative of \(\widetilde{\varphi}^{(c)}\), as follows:  
\[\frac{d^{n}\widetilde{\varphi}^{(c)}}{d^{n}\lambda_{i}^{(c)}(k)}=-\frac{log_ {2}(U^{(c)}(k))B_{n}^{Ruary}n_{0}B_{n}^{Ruary}(P^{F}(k)H_{i}^{w,(c)}(k))^{2}}{ \ln 2(\sum\limits_{j\in\Gamma\cup\{i\}}\lambda_{i}^{(c)}(k)P^{F}(k)H_{i}^{w, (c)}(k)+n_{0}B_{n}^{Ruary})^{2}} \tag{69}\]  
From the constraints on the variables and the physical significance of the constraints, it can be seen for the second-order derivative that \(d^{n}\widetilde{\varphi}^{(c)}\)\(/\)\(d^{n}\lambda_{i}^{(c)}(k)\)\(\leq 0\); then, the objective function is a concave function. Also, the restriction constraints satisfy the convex set property, so the optimization problem is convex, as proven. Therefore problem \(C_{9}\) can be solved with CVX toolkit to obtain a global suboptimal solution for \(\lambda^{(c)}(k)\). In summary, because of the solver employs mixed-integer computation and the interior-point method to solve each subproblem, the computational complexity of the joint optimization of the relay drone communication rate is \(O((WIV)^{3})\), where \(W\) is the total number of iterations of the algorithm, \(I\) is the maximum number of iterations for the subproblems, and \(V\) is the maximum dimension of the decision variables. The algorithm execution process is shown in Algorithm 2. ```
0: Use the coordinates of the cluster drones, the relay drone, the jammer, the ground station, and constraints such as the BER threshold to calculate the initial feasible solutions \(P^{R,(0)}\), \(Q^{(0)}\), \(\lambda^{(0)}\), and \(U^{(0)}\).