# Multi-Objective Optimization for EE-SE Tradeoff in Space-Air-Ground Internet of Things Networks
## 3 Problem Solution
### Sub-Problem 2: Power Allocation  
The second sub-problem is power allocation, including transmit power of IoT devices and UAVs. For the given \(\mathbf{D}\) and \(\mathbf{A}\), power allocation sub-problem can be described as  
\[ \begin{split}\mathcal{SP}_{3}:&\quad\max_{\mathbf{P }_{i},\mathbf{P}_{U}}\frac{\sum_{u}r_{us}}{\sum_{i}p_{i}+\sum_{u}p_{u}+Up_{h}} \\ \text{s.t.}&\quad C4,C5,C6,C7,C9.\end{split}  \tag{15}\]  
This is a non-convex problem because objective function and constraint \(C7\) are non-convex. For \(C7\), \(r_{us}\) is a concave function of \(p_{u}\), so we can obtain its upper bounder at \(p_{u}^{0}\) through first-order Taylor expansion  
\[r_{us}\leq Wlog_{2}\bigg{(}1+\frac{p_{u}^{0}g_{us}}{WN_{1}}\bigg{)}+\frac{N_{ 1}W}{\big{(}N_{1}W+p_{u}^{0}\big{)}\ln 2}\Big{(}p_{u}-p_{u}^{0}\Big{)}. \tag{16}\]  
Denote the upper bounder as \(\bar{r}_{us}\), which is linear respect to \(p_{u}\). Constraint \(C7\) can be described as  
\[C7^{\prime}:\quad\sum_{i}r_{iu}\geq r_{us},\quad\forall u, \tag{17}\]  
which is convex. For the objective function, which is in fractional form, Dinkelbach algorithm is adopted. Introduce a non-negative parameter \(\lambda\), which is  
\[\lambda=\frac{\sum_{u}r_{us}}{\sum_{i}p_{i}+\sum_{u}p_{u}+Up_{h}}. \tag{18}\]The objective function can be transformed into a parametric subtractive form as  
\[ \begin{split} F(\lambda)&=\max_{\mathbf{P}_{i},\mathbf{P} _{il}}\sum_{u}r_{us}-\lambda(\sum_{i}p_{i}+\sum_{u}p_{u}+Up_{\mathrm{h}})\\ &=\max_{\mathbf{P}_{i},\mathbf{P}_{il}}R-\lambda P,\end{split}  \tag{19}\]  
which is continuous and strictly monotonic decreasing in \(\lambda\) and has an unique root. Denote \(\lambda^{*}\) is the the root of \(F(\lambda)\), which can be derived using the Dinkelbach method. Substituting (19) for the objective function of \(\mathcal{SP}_{3}\), the optimal solution set of the problem is the same as that of \(\mathcal{SP}_{3}\) with \(\lambda=\lambda^{*}\). According to the above derivation, the transformed sub-problem can be obtained as  
\[ \begin{split}\mathcal{SP}_{4}:&\max_{\mathbf{P}_{ i},\mathbf{P}_{il}}R-\lambda P\\ \text{s.t.}& C4,C5,C6,C7^{\prime},C9.\end{split}  \tag{20}\]  
Containing two variables, problem \(\mathcal{SP}_{4}\) can be solved by Lagrange dual algorithm because it is a convex problem. The Lagrange function of (20) can be described as  
\[ \begin{split}&\mathcal{L}(\mathbf{P}_{i},\mathbf{P}_{il},\mathbf{\mu}, \mathbf{\upsilon},\mathbf{\omega},\mathbf{\varepsilon},\mathbf{\tau})\\ &=R-\lambda P+\sum_{i}\mu_{i}(p_{\mathrm{imax}}-p_{i})\\ &+\sum_{u}v_{u}(p_{\mathrm{umax}}-p_{u})\\ &+\sum_{u}\omega_{u}(r_{\mathrm{us}}-r_{\mathrm{min}})\\ &+\sum_{u}\varepsilon_{u}\left(\sum_{i}r_{iu}-p_{us}\right)\\ &+\tau\left(\frac{\sum_{u}W\log_{2}\left(1+\frac{p_{u}\mathcal{P} _{us}}{WN_{1}}\right)}{U(W+BK)}-\varepsilon\right),\end{split}  \tag{21}\]  
where \(\mathbf{\mu},\mathbf{\upsilon},\mathbf{\omega},\mathbf{\varepsilon},\mathbf{\tau}\) are Lagrangian multipliers related to the constraints. The Lagrangian dual expression is  
\[ \begin{split}&\mathcal{G}(\mathbf{\mu},\mathbf{\upsilon},\mathbf{\omega}, \mathbf{\varepsilon},\mathbf{\tau})\\ &=\sup_{\mathbf{P}_{i},\mathbf{P}_{il}}\mathcal{L}(\mathbf{P}_{i},\mathbf{P}_{il},\mathbf{\mu},\mathbf{\upsilon},\mathbf{\omega},\mathbf{\varepsilon},\mathbf{\tau }).\end{split}  \tag{22}\]  
Therefore, the Lagrangian dual problem can be described as  
\[ \begin{split}\mathcal{SP}_{5}:&\min_{(\mathbf{\mu},\mathbf{ \upsilon},\mathbf{\omega},\mathbf{\varepsilon},\mathbf{\tau})}\mathcal{G}(\mathbf{\mu},\mathbf{ \upsilon},\mathbf{\omega},\mathbf{\varepsilon},\mathbf{\tau})\\ \text{s.t.}&\mathbf{\mu},\mathbf{\upsilon},\mathbf{\omega},\mathbf{ \varepsilon},\mathbf{\tau}\succeq 0.\end{split}  \tag{23}\]  
Next, we derive the optimal solution with given \(\mathbf{\mu},\mathbf{\upsilon},\mathbf{\omega},\mathbf{\varepsilon},\mathbf{\tau}\). Due to \(C4\), the optimal solution \(p_{i}^{*}\) should satisfy the condition  
\[\left\{