# Multi-Objective Optimization for EE-SE Tradeoff in Space-Air-Ground Internet of Things Networks
## 2 System Model and Problem Formulation
### Problem Formulation  
Firstly, we provide expressions for energy efficiency and spectral efficiency, and construct the objective function for the optimization problem. Energy efficiency \(\eta_{\mathrm{EE}}\) is the ratio of effective information transmission rate to power. Since the data rate of the system depends on the communication rate between UAVs and the satellite, the EE can be expressesd as  
\[\eta_{\mathrm{EE}}=\frac{\sum_{u}r_{us}}{\sum_{i}p_{i}+\sum_{u}p_{u}+Up_{h}}, \tag{9}\]  
where \(p_{h}\) is the power required for each UAV to maintain levitation.  
The spectrum efficiency \(\eta_{\rm SE}\) is defined as the information rate divided by the bandwidth of the communication channel, which can be expressed as  
\[\eta_{\rm SE}=\frac{\sum_{u}r_{us}}{U(W+BK)}=\frac{\sum_{u}W\log_{2}\left(1+\frac {p_{u}\delta_{\rm uu}}{WN_{1}}\right)}{U(W+BK)}. \tag{10}\]  
For notational convenience, the horizontal and vertical coordinates of IoT device \(i\) and UAV \(u\) are denoted as \(\mathbf{d}_{i}^{\rm I}=(x_{i}^{\rm I},y_{i}^{\rm I})\) and \(\mathbf{d}_{u}^{\rm U}=(x_{u}^{\rm U},y_{u}^{\rm U})\). Letting \(\mathbf{D}=\{\mathbf{d}_{u}^{\rm U}\}\), \(\mathbf{A}=\{a_{iu}\}\), \(\mathbf{P}_{I}=\{p_{i}\}\) and \(\mathbf{P}_{U}=\{p_{u}\}\), the problem can be described as  
\[\mathcal{P}_{1}: \max_{\mathbf{A},\mathbf{P}_{1},\mathbf{P}_{U},\mathbf{D}}\eta_{ \rm EE},\eta_{\rm SE}\] (11) s.t. \[\mathrm{C1}:a_{iu}\in\{0,1\},\quad\forall i,u,\] \[\mathrm{C2}:\sum_{i}a_{iu}=K,\quad\forall u,\] \[\mathrm{C3}:\sum_{u}a_{iu}\leq 1,\quad\forall i,\] \[\mathrm{C4}:0\leq p_{i}\leq p_{\rm imax},\quad\forall i,\] \[\mathrm{C5}:0\leq p_{u}\leq p_{\rm umax},\quad\forall u,\] \[\mathrm{C6}:r_{iu}\geq r_{\rm min},\quad\forall i,\] \[\mathrm{C7}:\sum_{i}r_{iu}\geq r_{us},\quad\forall u,\] \[\mathrm{C8}:||\mathbf{d}_{m}^{\rm U}-\mathbf{d}_{n}^{\rm U}||^{2 }\geq d_{\rm min},\quad\forall m,n\in\mathcal{U},m\
eq n,\]  
where \(r_{\rm min}\) represents the minimum rate requirement for IoT devices to transmit data. \(p_{\rm imax}\) and \(p_{\rm umax}\) represent the maximum transmitting power of IoT devices and UAVs respectively. \(d_{\rm min}\) represents the minimum distance between two UAVs.  
In problem \(\mathcal{P}_{1}\), constraint \(C1\) stipulates that \(a_{iu}\) is a binary variable. \(a=1\) means there is communication between the UAV and the IoT device, otherwise \(a=0\). Constraints \(C2\) and \(C3\) limit the number of communication devices. They ensure that each UAV can communicate with \(K\) IoT devices and each IoT device can communicate with at most one UAV respectively. Constraints \(C4\) and \(C5\) indicate that the transmission power of IoT devices and UAVs cannot exceed the specified maximum transmission power. Constraint \(C6\) guarantees the rate at which each IoT device transmits data is greater than the minimum rate required to meet the requirements of service quality. Constraint \(C7\) means the rate at which the IoT device transmits data to the UAV is greater than the rate at which the UAV transmits data to the satellite. Furthermore, constraint \(C8\) guarantees that there is adequate safety distance between two UAVs. This ensures that any UAVs will not collide with each other.