## 4 Methodology
### Overall Time Complexity Analysis of Algorithm  
The proposed method integrates dynamic programming, multi-objective optimization, as well as distributed gradient descent and Gradient Surgery techniques, aiming to optimize resource scheduling and task offloading in satellite networks. To comprehensively evaluate the time complexity of this approach, it is essential to analyze the complexity of each component module.  
In the dynamic resource scheduling and task offloading framework, dynamic programming (DP) and value iteration techniques are employed to update system states and decision variables in real time, thereby optimizing resource allocation and task scheduling strategies. Assuming the size of the system state space is \(|S|\), and the size of the action space is \(|A|\), the value iteration process requires \(T\) iterations, with each iteration having a time complexity of \(O(|S|\times|A|)\). Therefore, the total time complexity of dynamic programming and value iteration is \(O(T\times|S|\times|A|)\). Additionally, task offloading decisions are represented by a discrete matrix \(\mathbf{X}(t)\). Considering the number of tasks \(M\) and the number of nodes \(N\), the time complexity for updating \(\mathbf{X}(t)\) at each time step is \(O(N\times M)\).  
In the distributed optimization segment, this study employs multi-objective optimization and distributed gradient descent algorithms to balance the minimization of latency and maximization of resource utilization. Assuming the number of objective functions is \(K\) (typically a constant, such as 2), gradient descent requires \(I\) iterations, with each iteration necessitating the computation of gradients for each task and node, resulting in a time complexity of \(O(K\times N\times M\times I)\). Gradient Surgery is utilized to adjust the gradient directions to ensure that the gradients do not conflict during the multi-objective optimization process, with a time complexity of \(O(K^{2}\times N\times M)\). Since Gradient Surgery is performed during each gradient update, its overall time complexity is proportional to the number of gradient descent iterations \(I\). Thus, the total time complexity of distributed gradient descent and Gradient Surgery is \(O(K^{2}\times N\times M\times I)\).  
The overall time complexity of the proposed method, which combines the complexities of the aforementioned components, can be approximated as \[O(T\times|S|\times|A|+K^{2}\times N\times M\times I)\]  
Considering this, in practical applications, the state space \(|S|\) and action space \(|A|\) are typically related to the number of nodes \(N\) and tasks \(M\) (e.g., \(|S|=O(N\times M)\) and \(|A|=O(N)\)); the above time complexity can be further simplified to  
\[O(T\times N^{2}\times M+K^{2}\times N\times M\times I)\]  
where \(T\) is the number of value iterations, \(N\) is the number of nodes, \(M\) is the number of tasks, \(K\) is the number of objective functions, and \(I\) is the number of gradient descent iterations.  
Moreover, although distributed optimization techniques theoretically allow for the distribution of time complexity across multiple computing nodes, thereby reducing the computational burden on individual nodes, in practice, communication overhead and synchronization delays are significant factors that must be considered. These factors typically introduce additional time complexity, specifically \(O(N\times M)\) for communication overhead. Therefore, while distributed optimization offers substantial advantages in parallel computation, the actual time performance requires balancing computational and communication overheads.