# 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
For RSS system \(n\), by dropping the constant term in Equation (32), which does not affect the solution, we can obtain the subproblem to be solved by the RSS system \(n\in N\) in the \((t+1)\)th iteration  
\[\mathrm{P5}:\min-\underset{1\leq i\leq|OM_{\mathrm{R}}|}{\sum}z_{n,i}+ \underset{s^{R}_{n,i,j}\in A^{R}_{RL}}{\sum}\left(\lambda_{n}^{t}(sr^{k}_{n,i,j})\hat{g}_{n}(sr^{k}_{n,i,j})+\frac{\rho}{2}\left(\hat{g}_{n}(sr^{k}_{n,i,j} )-y^{t}(sr^{k}_{n,i,j})\right)^{2}\right)\] \[\mathrm{s.t.}\quad\left\{\mathbf{z}_{n},\mathbf{x}_{n},\mathbf{y}_{n}^{L}, \mathbf{\hat{g}}_{n}^{C}\right\}\in\xi_{n}\]  
Similarly, the subproblem to be solved by the DRS system in the \(t+1\) iteration is given by  
\[\mathrm{P6}:\min\sum\limits_{s^{R}_{n,i,j}\in A_{RL}}\left( \lambda_{r}^{t}(sr^{k}_{n,i,j})\hat{g}_{r}(sr^{k}_{n,i,j})+\frac{\rho}{2} \left(\hat{g}_{r}(sr^{k}_{n,i,j})-y^{t}(sr^{k}_{n,i,j})\right)^{2}\right)\] \[\mathrm{s.t.}\quad\mathbf{\hat{g}}_{r}^{C}\in\xi_{r}\]  
On account of the quadratic objective and convex feasible set of problems P5 and P6, both the problems are convex optimization problems that can be solved directly by applying classical methods such as the primal-dual interior-point method [28] or optimization software such as CVX. **Step 2. Updating Global Variables:**  
After the update of local variables, the global variables are updated by minimizing the augmented Lagrangian in each iteration, which is given as follows  
\[\mathbf{y}^{C,t+1}=\underset{y^{C}}{\arg}\min\left\{ \begin{array}{l}\sum\limits_ {sr^{k}_{n,i,j}\in A_{RL}}{\lambda_{n}^{t}(sr^{k}_{n,j,j})\left(\hat{y}_{n}^{ t+1}(sr^{k}_{n,i,j})-y(sr^{k}_{n,i,j})\right)}\\ +\underset{sr^{k}_{n,i,j}\in A_{RL}}{\sum}\lambda_{r}^{t}(sr^{k}_{n,i,j}) \left(\hat{y}_{r}^{t+1}(sr^{k}_{n,i,j})-y(sr^{k}_{n,i,j})\right)\\ +\frac{\rho}{2}\underset{sr^{k}_{n,i,j}\in A_{RL}}{\sum}\left(\hat{y}_{n}^{t+ 1}(sr^{k}_{n,i,j})-y(sr^{k}_{n,i,j})\right)^{2}\\ +\frac{\rho}{2}\underset{sr^{k}_{n,i,j}\in A_{RL}}{\sum}\left(\hat{y}_{r}^{t+ 1}(sr^{k}_{n,i,j})-y(sr^{k}_{n,i,j})\right)^{2}\end{array} \right\} \tag{34}\]  
Since Equation (34) is an unconstrained quadratic programming problem, which is strictly convex owing to the additional quadratic regularization terms, the global variable \(\mathbf{y}^{C,t+1}\) can be solved by setting its gradient to 0, i.e.,  
\[\rho\left(\hat{y}_{n}^{t+1}(sr^{k}_{n,i,j})+\hat{y}_{r}^{t+1}(sr^{k}_{n,i,j})-2 y(sr^{k}_{n,i,j})\right)+\lambda_{n}^{t}(sr^{k}_{n,i,j})+\lambda_{r}^{t}(sr^{k}_{n,i,j})=0, \ \forall sr^{k}_{n,i,j}\in A_{RL} \tag{35}\]Then we can obtain  
\[y^{t+1}(sr^{k}_{n,i,j})=\frac{1}{2\rho}\Big{(}\lambda^{t}_{n}(sr^{k}_{n,i,j})+ \lambda^{t}_{r}(sr^{k}_{n,i,j})\Big{)}+\frac{1}{2}\Big{(}y^{t+1}_{n}(sr^{k}_{n, i,j})+\hat{y}^{t+1}_{r}(sr^{k}_{n,i,j})\Big{)},\ \forall sr^{k}_{n,i,j}\in A_{RL} \tag{36}\]  
By initializing the Lagrange multiplier as zero, the above equation can be reduced to  
\[y^{t+1}(sr^{k}_{n,i,j})=\frac{1}{2}\Big{(}\hat{y}^{t+1}_{n}(sr^{k}_{n,i,j})+ \hat{y}^{t+1}_{r}(sr^{k}_{n,i,j})\Big{)},\quad\forall sr^{k}_{n,i,j}\in A_{RL} \tag{37}\]  
That is, in each iteration, the global variable \(y^{t+1}(sr^{k}_{n,i,j})\) can be obtained by averaging the corresponding local copies of RSS system \(n\) and the DRS system. **Step 3. Updating Lagrange multiplier:**  
Applying the penalty parameter \(\rho\) as step size, the Lagrange multipliers are updated as  
\[\Big{\{}\lambda^{t+1}_{n}\Big{\}}_{1\leq n\leq N}= \lambda^{t}_{n}+\rho(\hat{y}^{C,t+1}_{n}-y^{C,t+1}_{n}) \tag{38}\]  
\[\lambda^{t+1}_{r}=\lambda^{t}_{r}+\rho(\hat{y}^{C,t+1}_{r}-y^{C,t+1}_{r}) \tag{39}\]  
Equations (38) and (39) imply that step 3 can be executed by \(N\) RSS systems and the DRS systems in parallel. More specifically, each RSS system updates its associate Lagrange multipliers by Equation (38), and the DRS system updates its associate Lagrange multipliers by Equation (39).