# Multi-Objective Optimization for EE-SE Tradeoff in Space-Air-Ground Internet of Things Networks
## 3 Problem Solution
### Sub-Problem 3: UAV Position Deployment
 \tag{38}\]  
For any \(\mathbf{t}=[t_{1},t_{2}]^{T}\), the following formula can be obtained  
\[\mathbf{t}^{T}\mathbf{H} =\frac{a_{iu}B}{x^{4}y^{4}\ln 2\bigg{(}C_{1}10^{\frac{C_{2}}{y}}+x^{2 }\bigg{)}} \tag{39}\] \[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\] \[+\bigg{(}C_{1}C_{2}^{2}10^{\frac{C_{2}}{y}}\ln^{2}10x^{4}+2C_{1} ^{2}C_{2}10^{\frac{2C_{2}}{y}}\ln 10x^{2}y\] \[+2C_{1}C_{2}10^{\frac{C_{2}}{y}}\ln 10x^{4}y\bigg{)}t_{2}^{2}\] \[+4C_{1}C_{2}10^{\frac{C_{2}}{y}}\ln^{2}10x^{3}y^{2}t_{1}t_{2}\bigg{]}\] \[\geq\frac{a_{iu}B}{x^{4}y^{4}\ln 2\bigg{(}C_{1}10^{\frac{C_{2}}{y}}+ x^{2}\bigg{)}}\] \[\cdot\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{2C_{2}}{y}}y^{4}+2C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\] \[+\bigg{(}2C_{1}^{2}C_{2}10^{\frac{2C_{2}}{y}}\ln 10x^{2}y+2C_{1} C_{2}10^{\frac{C_{2}}{y}}\ln 10x^{4}y\bigg{)}t_{2}^{2}\] \[+\bigg{(}2\sqrt{C_{1}10^{\frac{C_{2}}{y}}}xy^{2}t_{1}+\sqrt{C_{1} 10^{\frac{C_{2}}{y}}}C_{2}\ln 10x^{2}t_{2}\bigg{)}^{2}\bigg{]}\] \[\geq 0.\]  
According to (39), \(r_{iu}\) is a convex function of \(x,y\). Therefore, according to the [PERSON] expansion, \(r_{iu}\) at point \(x^{0},y^{0}\) can be lower-bounded by  
\[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\]0 \[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\]1 \[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\]2 \[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\]3 \[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\]4  
where \(\Psi\) and \(\Omega\) are the partial derivatives of \(r_{iu}\) against \(x\) and \(y\), respectively, which can be expressed as  
\[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\]5  
\[\times\bigg{[}\bigg{(}2C_{1}^{2}10^{\frac{C_{2}}{y}}y^{4}+6C_{1} 10^{\frac{C_{2}}{y}}x^{2}y^{4}\bigg{)}t_{1}^{2}\]6For the non-convex constraints \(\mathcal{C}\mathcal{P}^{\prime}\), we can know it is a convex function of \(\|\mathbf{d}_{u}-\mathbf{d}_{i}\|\). Therefore, \(\theta_{i}u\) at point \(\mathbf{d}_{u}^{0}\) can be lower-bounded by  
\[ \begin{split}\theta_{iu}&\geq\arctan\left(\frac{h_{ \text{U}}}{\|\mathbf{d}_{u}^{\text{U}}-\mathbf{d}_{i}^{\text{I}}\|}\right)\\ &+\Phi(\mathbf{d}_{u}^{\text{U0}})\left(\|\mathbf{d}_{u}^{\text{U} }-\mathbf{d}_{i}^{\text{I}}\|-\|\mathbf{d}_{u}^{\text{U0}}-\mathbf{d}_{i}^{ \text{I}}\|\right)\\ &=\bar{\theta}_{iu},\quad\forall i,u,\end{split}  \tag{43}\]  
where  
\[\Phi=\frac{\partial\theta_{iu}}{\partial\|\mathbf{d}_{u}^{\text{U}}-\mathbf{d }_{i}^{\text{I}}\|}=-\frac{h_{\text{U}}}{\|\mathbf{d}_{u}^{\text{I}}-\mathbf{ d}_{i}^{\text{I}}\|^{2}+h_{\text{U}}^{2}}. \tag{44}\]  
According to (35), (40), and (43), the problem can be described as  
\[ \begin{split}\mathcal{S}\mathcal{P}_{9}:&\max_{ \mathbf{D},\Theta}\sum_{i}\bar{r}_{iu}\\ \text{s.t.}& C\mathcal{C}^{\prime}:\bar{r}_{iu}\geq r _{\min},\quad\forall i,\\ & C\mathcal{T}^{\prime\prime}:\bar{r}_{iu}\geq r_{us},\quad \forall u,\end{split}  \tag{45}\]  
which is a convex optimization problem. So far, we have obtained the solutions of the three sub-problems. On this basis, we jointly optimize the sub-channel selection, power allocation, and UAV position deployment through the BCD process. First, for a given suitable \(\mathbf{D}^{0}\), \(\mathbf{P}_{I}^{0}\), and \(\mathbf{P}_{U}^{0}\), solve sub-problem 1 to obtain \(\mathbf{A}^{1}\). Then, according to the obtained \(\mathbf{A}^{1}\) and fixed \(\mathbf{D}^{0}\), solve sub-problem 2 to obtain \(\mathbf{P}_{I}^{1}\) and \(\mathbf{P}_{U}^{1}\), and then solve sub-problem 3 to obtain \(\mathbf{D}^{1}\).