## 4 Joint Message-Passing and Convex Optimization Framework for UAV Scheduling
### Design Concepts and Contributions  
The proposed energy-efficient UAV-based surveillance system is designed based on the following concepts.  
* First, MWIS-based scheduling is conducted to select UAVs whose cameras can be turned off when the corresponding target areas are monitored by other UAVs (i.e., visually overlapping areas). According to this scheduling, the operational energy consumption for the selected UAVs can be reduced and their surveillance lifetime can be extended. In order to solve this MWIS-based scheduling problem, message-passing is used in this paper because it is a well-known solution for this type of combinatorial problem. More details are in Section 4.2.1.
* Furthermore, the unscheduled UAVs are moved to the charging towers to be charged with wireless energy transfer technologies. Subsequently, our proposed optimization framework determines the charging matching of UAVs to charging towers. In addition, each tower determines how much energy should be delivered to the matched UAVs in order to conduct this matching decision in each unit time. This formulation is a mathematically non-convex optimization; however, we converted the corresponding non-convex terms into convex terms (i.e., the polynomial-time operation can be realized for the given joint matching and charging optimization formulation). More details are in Section 4.2.2.  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Notation** & **Value** \\ \hline Aircraft weight including battery and propellers, \(W\) & 1375 g \\ Rotor radius, \(R\) & 0.4 m \\ Rotor disc area, \(A=\pi R^{2}\) & 0.503 m\({}^{2}\) \\ Number of blades, \(b\) & 4 \\ Rotor solidity, \(s=\frac{0157b}{\pi R}\) & 0.05 \\ Blade angular velocity, \(\Omega\) & 300 radius/s \\ Tip speed of the rotor blade, \(U_{tip}=\Omega R^{2}\) & 120 \\ Fuselage drag ratio, \(d_{0}=\frac{0.0151}{sA}\) & 0.6 \\ Air density, \(\rho\) & 1.225 kg/m\({}^{3}\) \\ Mean rotor-induced velocity in hovering, \(v_{0}=\sqrt{\frac{W}{spA}}\) & 4.03 \\ Profile drag coefficient, \(\delta\) & 0.012 \\ Incremental correction factor to induced power, \(k\) & 0.1 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Energy-related notation and setup parameters [66].  
As discussed above, our first problem aims at the minimization of energy consumption by selecting certain amounts of UAVs those cameras can be turned off when the observing target areas are also monitored by other UAVs (visually overlapping areas). Therefore, the objective of this first problem is about to extend surveillance lifetime by saving energies in UAVs. In addition, the second problem is about to conduct joint optimization for charging matching and energy allocation decision between charging towers and UAVs based on the results of the first problem. The objective of this second problem is to maximize the energy charging amounts to the UAVs.  
For the given first problem, the UAVs whose cameras are turned off can be charged for extending surveillance lifetime. The charging should be conducted in an efficient way; thus, the second problem, which works based on the result of the first problem, is required for the energy charging amount maximization, which can be beneficial in terms of surveillance lifetime extension, eventually.