# Multi-Objective Optimization for EE-SE Tradeoff in Space-Air-Ground Internet of Things Networks
## 3 Problem Solution
### Sub-Problem 2: Power Allocation
\begin{split}\frac{\partial\mathcal{L}}{\partial p_{i}} \bigg{|}_{p_{i}=0}<0,\quad p_{i}^{*}=0,\\ \frac{\partial\mathcal{L}}{\partial p_{i}}\bigg{|}_{p_{i}=p_{i}^{* }}=0,\quad p_{i}^{*}>0.\end{split} \right. \tag{24}\]Similarly, the optimal solution \(p_{u}^{*}\) should satisfy the condition  
\[\left\{ \begin{aligned} \frac{\partial\mathcal{L}}{\partial p_{u}} \Big{|}_{p_{u}=0}<0,\quad p_{u}^{*}=0,\\ \frac{\partial\mathcal{L}}{\partial p_{u}}\Big{|}_{p_{u}=p_{u}^{*} }=0,\quad p_{u}^{*}>0.\end{aligned} \right. \tag{25}\]  
Because the Lagrangian dual function in (20) may be non-differentiable, the subgradient method is used to update the Lagrangian multipliers. We can update the Lagrangian multipliers \(\mathbf{\mu}\), \(\mathbf{\upsilon}\), \(\mathbf{\omega}\), \(\mathbf{\varepsilon}\), \(\mathbf{\tau}\) as follows  
\[\mu_{i}^{n+1}=[\mu_{i}^{n}-\gamma_{1}^{n}(p_{\text{imax}}-p_{i})]^{+}, \tag{26}\]  
\[v_{u}^{n+1}=[v_{u}^{n}-\gamma_{2}^{n}(p_{\text{umax}}-p_{u})]^{+}, \tag{27}\]  
\[\omega_{u}^{n+1}=[\omega_{u}^{n}-\gamma_{3}^{n}(r_{us}-r_{\text{min}})]^{+}, \tag{28}\]  
\[\varepsilon_{u}^{n+1}=\left[\varepsilon_{u}^{n}-\gamma_{4}^{n}\left(\sum_{i}r _{iu}-\tilde{r}_{us}\right)\right]^{+}, \tag{29}\]  
\[\tau^{n+1}=\left[\tau^{n}-\gamma_{5}^{n}\left(\frac{\sum_{u}W\log_{2}\left(1+ \frac{p_{u}S_{iu}}{WN_{1}}\right)}{U(W+BK)}-\epsilon\right)\right]^{+}, \tag{30}\]  
where \(n\) is the iterative index, \(\gamma\) is the step length, and \([\cdot]^{+}=\max\{0,\cdot\}\). We adopt the adaptive step size shown below  
\[\gamma_{i}^{n}=\frac{1}{\sqrt{n}},\quad i=1,2,3,4,5. \tag{31}\]  
In summary, Algorithm 2 gives the solution process of the power allocation sub-problem. ```
1:Input: Subchannel selection \(\mathbf{A}\), UAVs positon \(\mathbf{D}\), tradeoff coefficient \(\epsilon\), and the maximum number of iterations \(N\). 2: Initialize Lagrangian multipliers \(\mathbf{\mu}\), \(\mathbf{\upsilon}\), \(\mathbf{\omega}\), \(\mathbf{\varepsilon}\), \(\mathbf{\tau}\), variable \(\lambda\), and tolerance \(\Delta_{1}\), \(\Delta_{2}\). 3:while\(|F(\lambda_{k})|>\Delta_{1}\) and \(k\leq N\)do
4:while\(|\Phi^{n}|>\Delta_{2}\) and \(n\leq N\)do
5: Use (24) and (25) to obtain the optimal power \(\mathbf{P}_{I}^{n+1}\) and \(\mathbf{P}_{II}^{n+1}\). 6: Use (26)\(\sim\)(31) to obtain update multipliers \(\mathbf{\mu}^{n+1}\), \(\mathbf{\upsilon}^{n+1}\), \(\mathbf{\omega}^{n+1}\), \(\mathbf{\varepsilon}^{n+1}\), \(\mathbf{\tau}^{n+1}\). 7: Calculate the difference \(\Phi^{n}=\frac{\|\mathbf{\mu}^{n+1}-\mathbf{\mu}^{n}\|}{\|\mathbf{\mu}^{n}\|}-\frac{\|\mathbf{ \
u}^{n+1}-\mathbf{\
u}^{n}\|}{\|\mathbf{\upsilon}^{n}\|}-\frac{\|\mathbf{\omega}^{n+1}- \mathbf{\omega}^{n}\|}{\|\mathbf{\alpha}^{n}\|}-\frac{\|\mathbf{\varepsilon}^{n+1}-\mathbf{ \sigma}^{n}\|}{\|\mathbf{\varepsilon}^{n}\|}-\frac{\|\mathbf{\tau}^{n+1}-\mathbf{\tau}^{n} \|}{\|\mathbf{\tau}^{n}\|}\). 8:\(n=n+1\). 9:endwhile
10:\(\lambda_{k+1}\leftarrow\frac{R}{p}\) with \(\mathbf{P}_{I}^{k}\) and \(\mathbf{P}_{U}^{k}\). 11:\(k=k+1\). 12:endwhile
13:Input: Power allocation \(\mathbf{P}_{I}\) and \(\mathbf{P}_{U}\). ```  
**Algorithm 2** Dinkelbach algorithm based on Lagrangian duality.