## 4 Methodology
### Multi-Objective Task Scheduling Based on Gradient Descent
The gradients of the objective functions concerning the decision variables are defined as follows:  
\[\
abla\mathbf{x}f_{1}(\mathbf{X})=\alpha\left(\delta_{ik}^{j}\right)_{i,j} \tag{31}\]  
\[\
abla\mathbf{x}f_{2}(\mathbf{X})=\beta\left(c_{i}+s_{i}+b_{i}\right)_{i,j} \tag{32}\]  
\[\
abla\mathbf{R}f_{2}(\mathbf{R})=\alpha\left(c_{i}+s_{i}+b_{i}\right) \tag{33}\]  
The combined gradient for the task offloading matrix is  
\[\
abla\mathbf{X}=\
abla\mathbf{x}f_{1}(\mathbf{X})+\
abla\mathbf{x}f_{2}( \mathbf{X})=\alpha\left(\delta_{ik}^{j}\right)_{i,j}+\beta\left(c_{i}+s_{i}+b_ {i}\right)_{i,j} \tag{34}\]  
The update rule for \(\mathbf{X}\) is given by  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]0  
where \(\eta\) is the learning rate. This iterative process drives the task offloading matrix towards an optimal solution by moving in the direction of the gradient of the objective function. However, since the two objectives \(f_{1}(\mathbf{X})\) and \(f_{2}(\mathbf{X})\) may conflict, directly applying gradient descent could lead to suboptimal solutions. To mitigate this, Gradient Surgery is employed to adjust the gradient directions, ensuring that the updates balance both the latency minimization and resource utilization maximization. The adjusted gradients are defined as  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]1  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]2  
where \(\
abla_{\mathbf{R}}\) is the gradient related to resource allocation. These adjusted gradients ensure that the optimization process moves towards a Pareto optimal solution, where no objective can be improved without worsening another. The framework optimizes task offloading decisions and resource allocation plans through iterative adjustments using distributed gradient descent and Gradient Surgery. This approach ensures that the system converges to an optimal policy, which maximizes long-term resource utilization while minimizing task delays and thereby enhancing the overall performance and robustness of the satellite network. Additionally, the interdependence between task offloading and computational resource allocation plays a crucial role in optimizing both system latency and resource utilization. The task offloading matrix \(\mathbf{X}\) determines the allocation of tasks to nodes, while the resource allocation matrix \(\mathbf{R}\) defines the computational resources allocated to each task at each node. Specifically, for task \(\mathcal{T}_{j}\) offloaded to node \(\mathcal{S}_{i}\), the computational demand \(t_{j}^{c}\) must be satisfied by the allocated resources \(R_{i}^{j}\), i.e.,  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]3  
This interdependence leads to a coupled multi-objective optimization problem:  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]4  
s.t. \[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]5  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]6  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]7  
In the presence of multiple objectives, Gradient Surgery is employed to adjust the gradients for task offloading and resource allocation, ensuring that the update direction balances latency and resource utilization. The combined gradient for the task offloading matrix is  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]8  
The adjusted gradients ensure that the optimization process proceeds towards a Pareto optimal solution, where both latency and resource consumption are optimized without conflicting gradients.