# Design of an Integrated System for Spaceborne SAR Imaging and Data Transmission
## 2 Construction of Mathematical Models
### Algorithms  
#### 2.3.1 Alternating Direction Method of Multipliers  
The alternating direction method of multipliers (ADMM) is a highly efficient optimization technique, especially for decomposing large-scale problems. By utilizing alternating optimization and updating Lagrange multipliers, ADMM can attain the optimal solution. It is particularly effective for addressing problems with a separable structure, such as those that can be divided into two or more subproblems [34]. To solve the power allocation problem, we utilize the ADMM and penalty function constraint methods. By keeping other parameters constant, we convert the original non-convex problem into a convex one. The binary constraint (12.d) is transformed into the following form:  
\[u_{k}(1-u_{k})\leq 0 \tag{13}\]  
The optimization problem \(\mathcal{P}_{1}\) is transformed into:  
\[ \begin{split}&(\mathcal{P}_{2})\underset{\boldsymbol{u}, \boldsymbol{p}}{\max}\;SNR_{\text{s}}(\boldsymbol{u},\boldsymbol{p})\\ &\text{s.t.}\left\{\begin{array}{ll}\sum\limits_{k=1}^{K}p_{k} \leq P&(14\;.a)\\ \xi\leq p_{k}\leq\frac{\eta P}{K}&(14\;.b)\\ \sum\limits_{k=1}^{K}\text{log}_{2}\bigg{(}1+\frac{|\alpha_{k}|_{2}(1-u_{n})p_{ k}}{\sigma_{\text{c}}^{2}}\bigg{)}\geq\kappa\;|\boldsymbol{u}|_{1}=\kappa\sum \limits_{k=1}^{K}u_{k}&(14\;.c)\\ u_{k}(1-u_{k})\leq 0&(14\;.d)\end{array}\right.\end{split}  \tag{14}\]  
Constraint (14.c) is a non-convex constraint. By converting it into a penalty function for the objective function, the optimization problem \(\mathcal{P}_{2}\) is transformed into:  
\[ \begin{split}&(\mathcal{P}_{3})\underset{\boldsymbol{u}, \boldsymbol{p}}{\max}\;\text{SNR}_{\text{s}}(\boldsymbol{u},\boldsymbol{p})+ \rho G(\boldsymbol{u},\boldsymbol{p})\\ &\text{s.t.}\left\{\begin{array}{ll}\sum\limits_{k=1}^{K}p_{k} \leq P&(15\;.a)\\ \xi\leq p_{k}\leq\frac{\eta P}{K}&(15\;.b)\\ u_{k}(1-u_{k})\leq 0&(15\;.c)\end{array}\right.\end{split}  \tag{15}\]  
where  
\[G(\boldsymbol{u},\boldsymbol{p})=\sum\limits_{k=1}^{K}\bigg{(}\text{log}_{2} \bigg{(}1+\frac{|\alpha_{k}|_{2}(1-u_{n})p_{k}}{\sigma_{\text{c}}^{2}}\bigg{)} -\kappa u_{k}\bigg{)} \tag{16}\]where \(\rho\) serves as the penalty factor that ensures the convergence of the algorithm, setting \(\rho=0.001\), ensuring that the objective function adheres to the given constraint. To simplify processing, the log2(\(\cdot\)) function is replaced with the log(\(\cdot\)) function. This modification does not affect the convergence results. In Equation (16), combining variables \(\mathbf{u}\) and \(\mathbf{p}\) within the log2(\(\cdot\)) function complicates the solution process. To resolve this issue, an additional variable \(\mathbf{w}\) is introduced. By using the method of Lagrange multipliers, the variables are successfully separated from the logarithmic function [35]. With fixed values of \(\mathbf{u}\) and \(\mathbf{p}\), the optimization problem \(\mathcal{P}_{3}\) can be transformed using the method of Lagrange multipliers as follows:  
\[(\mathcal{P}_{4})\underset{\mathbf{w}}{\text{max}}\sum_{k=1}^{K}\rho (\log(1+w_{k})) \tag{17}\] \[\text{s.t.} \begin{cases}w_{k}=\frac{|\alpha_{k}|_{2}(1-u_{n})p_{k }}{\sigma_{\text{c}}^{2}}&(\ref{eq:1}.\text{a})\\ \frac{|\alpha_{k}|_{2}\xi}{\sigma_{\text{c}}^{2}}\leq w_{k}\leq\frac{\eta P| \alpha_{k}|_{2}(1-u_{n})p_{k}}{K\sigma_{\text{c}}^{2}}&(\ref{eq:1}.\text{b}) \end{cases} \]  
The Lagrange multipliers for Equation (17) are expressed as follows:  
\[(\mathcal{L})(\mathbf{w},\mathbf{v})=\sum_{k=1}^{K}\rho(\ln(1+w_{k}))-v_{k}w_{k}+v_{k }\frac{|\alpha_{k}|_{2}(1-u_{k})p_{k}}{\sigma_{\text{c}}^{2}} \tag{18}\]  
\[w_{k}=min(max(\frac{1}{\
u}-1,\frac{|\alpha_{k}|_{2}\xi}{\sigma_{\text{c}}^{2 }}),\frac{\eta P|\alpha_{k}|_{2}(1-u_{k})p_{k}}{K\sigma_{\text{c}}^{2}}) \tag{19}\]  
\[v_{k}=v_{k}+\mu(\frac{|\alpha_{k}|_{2}(1-u_{k})p_{k}}{\sigma_{\text{c}}^{2}}- w_{k}) \tag{20}\]  
where \(\mathbf{v}\) represents the Lagrange multipliers and \(\mu\) denotes the penalty function used for the iterations in Equation (20).