## 4 Methodology
### Domain Clustering Based on Multi-Path Transmission
#### 4.2.6 Problem Modeling and Optimization Algorithm Design  
We address the satellite network resource allocation optimization using the Lagrange multiplier method and dual problem solving. The approach involves decomposing the main problem into sub-problems and applying the Karush-Kuhn-Tucker (KKT) conditions to ensure feasibility and solvability. The utility function for each user \(U_{l,m}\) is defined as the difference between its transmitted throughput and the minimum required throughput:  
\[U_{l,m}=\sum_{f=1}^{F}\sum_{t=1}^{T}\epsilon_{l,m,f,t}\log_{2}\left(1+\frac{P_ {l,m,f,t}}{S_{l,m,f,t}}\right)-R_{\min,l,m} \tag{44}\]  
where \(\epsilon_{l,m,f,t}\) is the interference coefficient for user \(l,m\) at frequency \(f\) and time \(t\). \(P_{l,m,f,t}\) is the power allocation for user \(l,m\) at frequency \(f\) and time \(t\). \(S_{l,m,f,t}\) is the signal-to-noise ratio for user \(l,m\) at frequency \(f\) and time \(t\). \(R_{\min,l,m}\) is the minimum acceptable throughput for user \(l,m\). The optimization objective is to maximize the total utility across all users:  
\[\max\sum_{l=1}^{L}\sum_{m=1}^{M}\sum_{f=1}^{F}\sum_{t=1}^{T}\epsilon_{l,m,f,t} \log_{2}\left(1+\frac{P_{l,m,f,t}}{S_{l,m,f,t}}\right)-R_{\min,l,m} \tag{45}\]  
s.t. \[P_{\min,l,m}\leq P_{l,m,f,t}\leq P_{\max,l,m},\quad\forall l,m,f,t\]  
\[\sum_{m=1}^{M}\sum_{t=1}^{T}\epsilon_{l,m,f,t}P_{l,m,f,t}S_{l,m,f,t}\leq I_{ \text{th},l,f},\quad\forall l,f\]  
\[S_{l,m,f,t}\geq S_{\text{NR},l,m},\quad\forall l,m,f,t\]  
To solve the optimization problem, we formulate the Lagrangian:  
\[ \begin{split} L(P,\lambda,\mu,\
u)&=\sum_{l=1}^{L} \sum_{m=1}^{M}\sum_{f=1}^{F}\sum_{t=1}^{T}\epsilon_{l,m,f,t}\log_{2}\left(1+ \frac{P_{l,m,f,t}}{S_{l,m,f,t}}\right)-R_{\min,l,m}\\ &\quad+\sum_{l=1}^{L}\sum_{f=1}^{F}\lambda_{l,f}\left(I_{\text{ th},l,f}-\sum_{m=1}^{M}\sum_{t=1}^{T}\epsilon_{l,m,f,t}P_{l,m,f,t}S_{l,m,f,t} \right)\\ &\quad+\sum_{l=1}^{L}\sum_{m=1}^{M}\sum_{f=1}^{F}\sum_{t=1}^{T} \mu_{l,m,f,t}\left(P_{\max,l,m}-P_{l,m,f,t}\right)\\ &\quad+\sum_{l=1}^{L}\sum_{m=1}^{M}\sum_{f=1}^{F}\sum_{t=1}^{T} \
u_{l,m,f,t}\left(S_{l,m,f,t}-S_{\text{NR},l,m}\right)\end{split}  \tag{46}\]where \(\lambda_{l,f}\), \(\mu_{l,m,f,t}\), and \(\
u_{l,m,f,t}\) are Lagrange multipliers corresponding to the interference threshold, maximum power limit, and signal-to-noise ratio constraints, respectively. The dual problem is then formulated as  
\[\min_{\lambda,\mu,\
u}\max_{P}L(P,\lambda,\mu,\
u) \tag{47}\]  
Applying the KKT conditions, we derive the necessary conditions for optimality:  
1. Primal Feasibility: All constraints must be satisfied. 2. Dual Feasibility: The Lagrange multipliers must be non-negative, i.e., \(\lambda_{l,f},\mu_{l,m,f,t}\), \(\
u_{l,m,f,t}\geq 0\). 3. Stationarity: The gradient of the Lagrangian with respect to \(P_{l,m,f,t}\) must be zero: \[\frac{\partial L}{\partial P_{l,m,f,t}}=\frac{\epsilon_{l,m,f,t}}{S_{l,m,f,t} \Big{(}1+\frac{P_{l,m,f,t}}{S_{l,m,f,t}}\Big{)}\ln 2}-\lambda_{l,f}\delta_{l,m,f,t}- \mu_{l,m,f,t}+\
u_{l,m,f,t}=0\] (48)  
These conditions collectively ensure that the solution obtained is optimal under the given constraints. In the optimization process, the Karush-Kuhn-Tucker (KKT) conditions are essential for identifying the optimal solution based on the given constraints. One situation arises when constraints are inactive, meaning that they do not affect the optimal solution. In such cases, the corresponding Lagrange multipliers are zero, i.e., \(\lambda_{i}=0\). This typically occurs when resources, such as power or computational capacity, are abundant and far exceed the required demand. In such scenarios, the inactive constraints do not influence the optimization. We can adopt the approach of excluding these inactive constraints from the optimization process to enhance computational efficiency, focusing only on the active constraints that directly affect resource allocation.