# Distributed Coordination of Space-Ground Multiresources for Remote Sensing Missions
## 3 Problem Transformation and Decomposition
### Problem Decomposition
\begin{array}{cc}-x(f)&v=s(f),f\in F_{n}\\ 0&v\in V-\{s(f),d(f)\},f\in F_{n}\\ x(f)&v=d(f),f\in F_{n}\end{array} \right.\] \[\text{C3}: \sum\limits_{f\in F_{n}}x(ss_{n,i}^{k},f)\leq\text{C}(ss_{n,i}^{k }),\quad\forall ss_{n,i}^{k}\in A_{S}^{n}\] \[\text{C4}: \sum\limits_{f\in F_{n}}x(sr_{n,i,j}^{k},f)\leq\text{C}(sr_{n,i,j}^ {k})\cdot\hat{g}_{n}(sr_{n,i,j}^{k}),\quad\forall sr_{n,i,j}^{k}\in A_{RL}^{n}\] \[\text{C5}: \sum\limits_{f\in F_{n,i}}x(os_{n,i,j}^{k},f)\leq\text{C}(os_{n,i, j}^{k})\cdot y(os_{n,i,j}^{k}),\quad\forall os_{n,i,j}^{k}\in A_{O}^{n}\] \[\text{C6}: \sum\limits_{f\in F_{n}}x(sg_{n,i,j}^{k},f)\leq\text{C}(sg_{n,i, j}^{k})\cdot y(sg_{n,i,j}^{k}),\quad\forall sg_{n,i,j}^{k}\in A_{GL}^{n}\] \[\text{C7}: \sum\limits_{j:s_{n,i,j}^{k}\in A_{RL}^{n}}\hat{g}_{n}(sr_{n,i,j}^{ k})\leq 1,\quad\forall 1\leq k\leq K,1\leq i\leq|S_{n}|\] \[\text{C8}: \sum\limits_{j:ks_{n,i,j}^{k}\in A_{O}^{n}}y(os_{n,i,j}^{k})\leq 1, \quad\forall 1\leq i\leq|OM_{n}|\] \[\text{C9}: \sum\limits_{is_{n,i,j}^{k}\in A_{O}^{n}}y(os_{n,i,j}^{k})\leq 1, \quad\forall 1\leq k\leq K,1\leq j\leq|S_{n}|\] \[\text{C10}: \sum\limits_{is_{n,i,j}^{k}\in A_{CL}^{n}}y(sg_{n,i,j}^{k})\leq 1, \quad\forall 1\leq k\leq K,1\leq j\leq|G_{n}|\] \[\text{C11}: \sum\limits_{j:ss_{n,i,j}^{k}\in A_{CL}^{n}}y(sg_{n,i,j}^{k})\leq 1, \quad\forall 1\leq k\leq K,1\leq i\leq|S_{n}|\]  
Similarly, the feasible solution set of the DRS system can be obtained  
\[\xi_{r}=\left\{\hat{g}_{r}^{C}\Bigg{|} \begin{array}{cc}\sum\limits_{n,i:sr_{ n,i,j}^{k}\in A_{RL}}\hat{g}_{r}(sr_{n,i,j}^{k})\leq 1,\quad\forall 1\leq k\leq K,1\leq j \leq|R|,\\ 0\leq\hat{g}_{r}(sr_{n,i,j}^{k})\leq 1,\quad\forall sr_{n,i,j}^{k}\in A_{RL} \end{array} \right\}. \tag{29}\]  
In order to further divide the coordinated resource scheduling problem with respect to the local variables, we define the local cost function for the resource scheduling of RSS system \(n\) as  
\[v_{n}(\mathbf{z}_{n},\mathbf{x}_{n},\mathbf{y}_{n}^{L},\hat{g}_{n}^{C})=\left\{ \begin{array} []{cc}-\sum\limits_{1\leq i\leq|OM_{n}|}z_{n,i},&\{\mathbf{z}_{n},\mathbf{x}_{n},\mathbf{y }_{n}^{L},\hat{g}_{n}^{C}\}\in\xi_{n}\\ +\infty,&\text{otherwise}\end{array} \right. \tag{30}\]Similarly, the local cost function for the DRS system is given by  
\[v_{r}(\mathbf{g}_{r}^{C})=\left\{ \begin{array}{ll}0,&\mathbf{\hat{g}}_{r}^{C}\in \xi_{r}\\ +\infty,&\text{otherwise}\end{array} \right. \tag{30}\]  
Based on Equations (27)-(30), P3 can be compactly rewritten as  
\[\text{P4}:\quad\min\sum\limits_{1\leq n<N}v_{n}(\mathbf{z}_{n},\mathbf{x }_{n},\mathbf{y}_{n}^{L},\mathbf{\hat{g}}_{C}^{C})+v_{r}(\mathbf{\hat{g}}_{r}^{C})\] \[\text{s.t.}\quad\hat{y}_{n}(sr_{n,ij}^{k})=y(sr_{n,ij}^{k}),\quad \forall sr_{n,ij}^{k}\in A_{RL}\] \[\quad\quad\quad\quad\quad\quad\hat{y}_{r}(sr_{n,ij}^{k})=y(sr_{n, ij}^{k}),\quad\forall sr_{n,ij}^{k}\in A_{RL}\]  
It can be seen that the objective of P4 is separable across the RSS and DRS systems, so that each system can handle the corresponding subproblem independently. Meanwhile, the consensus constraints guarantee the consistency of the local copies held in the RSS and DRS systems. As P4 is a typical global consistency optimization problem, it is solved in a distributed manner with the Alternating Direction Multiplier Method (ADMM) [25] in the next section.