# Distributed Coordination of Space-Ground Multiresources for Remote Sensing Missions
## 4 Distributed Coordinated Resource Scheduling Algorithm Design Based on ADMM
### Algorithm Implementation  
Algorithm 1 summarizes the ADMM-based distributed coordinated resource scheduling process of SGCN. To be specific, lines 3-7 are the implementation of iteration steps 1 to 3 and the associated information exchange between the RSS system and the DRS system. Line 10 indicates the termination criterion of the iteration, i.e., the DRS system informs each RSS system of the termination of the iteration when the global variables converge, where \(\varepsilon\) is a small positive number. In lines 11-12, the global variables \(\mathbf{y}^{C}\) and local variables \(\mathbf{y}^{L}_{n}\), \(\mathbf{z}_{n}\) are, respectively, recovered by the DRS system and RSS systems by employing Algorithms 2 and 3. Algorithm 2 is the recovery algorithm of the relaxed global variable \(\mathbf{y}^{C}\) for the DRS system. The main idea is to fix the global relaxed variables sequentially in descending order while avoiding conflicts with the already fixed variables. \(AC(sr^{k}_{n,i,j})\) denotes the set of link arcs in conflict with link arc \(sr^{k}_{n,i,j}\), which is defined as  
\[AC(sr^{k}_{n,i,j})=\Big{\{}sr^{k}_{n,i,j\text{ }}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!```
0: The mission requests and resource information of each satellite system. 0: Optimal resource scheduling results of each RSS system \(\{\mathbf{x}_{n}^{*},\mathbf{y}_{n}^{j},\mathbf{z}_{n}^{*}\}_{1\leq n\leq N}\). 1:Initialize:\(t\gets 0,\mathbf{y}^{C,t}\leftarrow\mathbf{0},\mathbf{\lambda}^{t}\leftarrow\mathbf{0}\);
2:repeat
3: Each RSS system solves problem P5 to update the local variables \(\{\mathbf{z}_{n}^{t+1},\mathbf{x}_{n}^{t+1},\mathbf{y}_{n}^{t+1},\mathbf{y}_{n}^{C,t+1}\}\);
4: DRS system solves problem P6 update local variables \(\mathbf{\hat{y}}_{n}^{C,t+1}\);
5: Each RSS system sends local copy variables \(\mathbf{\hat{y}}_{n}^{C,t+1}\) to the DRS system;
6: The DRS system updates the global variable \(\mathbf{y}^{C,t+1}\) through Equation (30), and distributes them to the corresponding RSS system;
7: Each RSS system updates the Lagrange multiplier \(\mathbf{\lambda}_{n}^{t+1}\) by Equation (31);
8: The DRS system updates the Lagrange multiplier \(\mathbf{\Lambda}_{r}^{t+1}\) through Equation (32);
9:\(t\gets t+1\);
10:until\(\|\mathbf{y}^{C+1}-\mathbf{y}^{C}\|_{2}\leq\varepsilon\)
11: DRS system executes Algorithm 2 to recover the relaxed variables \(\mathbf{y}^{C}\) and sends \((\mathbf{y}_{n}^{C})^{*}\) to each RSS system \(n\);
12: Each RSS system recovers relaxed variables \(\mathbf{y}_{n}^{L}\) and \(\mathbf{z}_{n}\) via Algorithm 3. ```

**Algorithm 1** Distributed coordinated resource scheduling algorithm based on the ADMM

Algorithm 3 is the recovery algorithm of the local relaxed variables \(\mathbf{y}_{n}^{L}\) and \(\mathbf{z}_{n}\) for each RSS system \(n\). Its main idea is to select a mission and allocate observation and transmission resources alternatively. Specifically, it first fixes the mission scheduling variable \(z_{n,\varepsilon}\) with the largest relaxed value. Then, it allocates observation and transmission resources to the fixed mission by finding a path on graph \(G_{R}(V_{R},A_{R})\), which is a subgraph of \(G_{K}\) related to RSS system \(n\). The value of the local relaxed variables associated with the observation arcs and the links arcs on the path can be fixed. Finally, update \(G_{R}(V_{R},A_{R})\) based on the resource allocation results and substitute the recovered variables into problem P5 and solve it again. The above process is repeated until all mission planning variables have been determined. The initial values of \(V_{R}\) and \(A_{R}\) are, respectively, defined as

\[ \begin{split} V_{R}&=V_{o}^{n}\cup V_{s}^{n}\cup V _{g}^{n}\cup V_{r}\cup V_{VT},\\ A_{R}&=A_{L}^{n}\cup A_{O}^{n}\cup A_{S}^{n}\cup A _{VT}^{n},\end{split}  \tag{41}\]

where \(V_{o}^{n}\), \(V_{s}^{n}\), and \(V_{g}^{n}\) denote the temporal replicas of the observation objects, RSSs, and ground stations of RSS system \(n\), respectively.