# Distributed Coordination of Space-Ground Multiresources for Remote Sensing Missions
## 4 Distributed Coordinated Resource Scheduling Algorithm Design Based on ADMM
### Augmented Lagrangian and ADMM Sequential Iterations  
Although the objective of P4 is in a separable form, its different parts are still coupled through the consistency constraint. In order to remove this obstacle to completely splitting P4, an augmented Lagrangian form is constructed for P4 [25; 26], which is given as follows  
\[ \begin{array}{ll}&L_{\rho}\big{(}\{\mathbf{z}_{n},\mathbf{x}_{n},\mathbf{y}_{n}^{L},\bm {\hat{g}}_{n}^{C}\}_{1\leq n\leq N},\{\mathbf{\hat{g}}_{r}^{C}\},\mathbf{y}^{C},\{ \mathbf{\lambda}_{n}\}_{1\leq n\leq N},\mathbf{\lambda}_{r}\big{)}\\ &=\sum\limits_{1\leq n\leq N}v_{n}(\mathbf{z}_{n},\mathbf{x}_{n},\mathbf{y}_{n}^{L},\mathbf{ \hat{g}}_{n}^{C})+v_{r}(\mathbf{\hat{y}}_{r})+\sum\limits_{sr_{n,ij}^{k}\in A_{RL }}\lambda_{n}(sr_{n,ij}^{k})\Big{(}\hat{y}_{n}(sr_{n,ij}^{k})-y(sr_{n,ij}^{k}) \Big{)}\\ &\quad+\sum\limits_{sr_{n,ij}^{k}\in A_{RL}}\lambda_{r}(sr_{n,ij}^{k})\Big{(} \hat{y}_{r}(sr_{n,ij}^{k})-y(sr_{n,ij}^{k})\Big{)}+\frac{\rho}{2}\sum\limits_ {sr_{n,ij}^{k}\in A_{RL}}\Big{(}\hat{y}_{n}(sr_{n,ij}^{k})-y(sr_{n,ij}^{k}) \Big{)}^{2}\\ &\quad+\frac{\rho}{2}\sum\limits_{sr_{n,ij}^{k}\in A_{RL}}\Big{(}\hat{y}_{r}( sr_{n,ij}^{k})-y(sr_{n,ij}^{k})\Big{)}^{2}\end{array}  \tag{31}\]  
where \(\mathbf{\lambda}_{n}=\Big{(}\{\lambda_{n}(sr_{n,ij}^{k})\}_{sr_{n,ij}^{k}\in A_{ RL}^{n}}\Big{)}\) and \(\mathbf{\lambda}_{r}=\Big{(}\{\lambda_{r}(sr_{n,ij}^{k})\}_{sr_{n,ij}^{k}\in A_{RL}} \Big{)}\) are the Lagrange multipliers with respect consistency constraints, and \(\rho\in\mathbb{R}_{++}\) is the penalty term, which is a constant parameter intended for adjusting the convergence speed of the ADMM [27]. Compared with the standard Lagrangian form, the additional quadratic regularization term in augmented Lagrangian can improve the convergence efficiency of the iterative method. Moreover, the additional quadratic regularization term is equal to zero for any feasible solution, i.e., the optimal solution of Equation (31) is equivalent to that of P4. As Equation (31) can be completely decomposed into \(N+1\) independent parts, P4 can be solved through sequential iterative local optimization for the RSS and DRS systems with the application of the ADMM. Let \(t\) denote the iteration index. The specific iteration steps are as follows:  
**Step 1. Updating Local Variables:**  
In each iteration, local variables are update by minimizing the augmented Lagrangian in Equation (31). Specifically, the RSS system \(n\) updates its local variables by \[\left\{\mathbf{z}_{n}^{t+1},\mathbf{x}_{n}^{t+1},\mathbf{y}_{n}^{L+1},\mathbf{\hat{g} }_{n}^{C,t+1}\right\}_{1\leq n\leq N} \tag{32}\] \[=\underset{\left\{\mathbf{z}_{n},\mathbf{x}_{n},\mathbf{y}_{n}^{t},\mathbf{\hat{g} }_{n}^{C}\right\}}{\arg}\min\left\{ \begin{array}{l}v_{n}(\mathbf{z}_{n},\mathbf{x}_{ n},\mathbf{y}_{n}^{L},\mathbf{\hat{g}}_{n}^{C})+\underset{s^{R}_{n,j,j}\in A^{R}_{RL}}{ \sum}\lambda_{n}^{t}(sr^{k}_{n,j,j})\left(\hat{g}_{n}(sr^{k}_{n,j,j})-y^{t}(sr ^{k}_{n,j,j})\right)\\ +\frac{\rho}{2}\underset{sr^{k}_{n,j,j}\in A^{R}_{RL}}{\sum}\left(\hat{g}_{n} (sr^{k}_{n,j,j})-y^{t}(sr^{k}_{n,j,j})\right)^{2}\end{array} \right\}\]  
Moreover, the DRS system updates its local variables via  
\[\hat{g}_{r}^{C,t+1}=\underset{\hat{g}_{r}^{C}}{\arg}\min\left\{ \begin{array}[] {l}v_{n}(\hat{g}_{r}^{C})+\underset{sr^{k}_{n,j,j}\in A_{RL}}{\sum}\lambda_{n }^{t}(sr^{k}_{n,j,j})\left(\hat{g}_{r}(sr^{k}_{n,j,j})-y^{t}(sr^{k}_{n,j,j}) \right)\\ +\frac{\rho}{2}\underset{sr^{k}_{n,j,j}\in A_{RL}}{\sum}\left(\hat{g}_{r}(sr^ {k}_{n,j,j})-y^{t}(sr^{k}_{n,j,j})\right)^{2}\end{array} \right\} \tag{33}\]  
It can be seen from Equations (32) and (33) that step 1 can be solved by the \(N\) RSS systems and the DRS system in parallel.