## 4 Methodology
### Multi-Objective Task Scheduling Based on Gradient Descent  
Due to bottleneck problems, traditional centralized computing methods struggle to solve resource scheduling issues in multi-node distributed environments due to bottleneck problems. To tackle this issue, we propose a task scheduling and resource management framework based on distributed optimization, integrating multi-objective optimization (MOML) and Gradient Surgery techniques. This framework leverages collaborative computing and edge computing to effectively enhance system performance, resolving resource conflicts and latency issues. Specifically, the framework employs a distributed gradient descent algorithm to optimize task offloading decisions and resource allocation schemes, utilizes multi-objective optimization methods to balance the objectives of minimizing latency and maximizing resource utilization, and coordinates the relationships between different optimization goals through Gradient Surgery techniques to ensure the convergence of the multi-objective optimization process. This approach not only improves resource utilization efficiency but also significantly reduces task processing latency. The resource scheduling and latency minimization problem in STIN can be modeled as a multi-objective optimization problem with the following objective function:  
\[\min_{\mathbf{X}}\left(\sum_{i=1}^{N}\sum_{j=1}^{M}\delta_{ik}^{j}X_{ij}+\sum _{i=1}^{N}\sum_{j=1}^{M}X_{ij}(c_{i}+s_{i}+b_{i})\right) \tag{25}\]  
Here, \(\delta_{ik}^{j}\) represents the transmission latency of task \(\mathcal{T}_{j}\) on satellite node \(\mathcal{S}_{i}\), while \(c_{i},s_{i}\), and \(b_{i}\) denote the computational, storage, and bandwidth resources of node \(\mathcal{S}_{i}\), respectively. The optimization problem can be decomposed into two separate objectives: minimizing latency and minimizing resource consumption. These are expressed as  
\[\min_{\mathbf{X}}f_{1}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{ j}}{b_{i}}X_{ij} \tag{26}\]  
\[\min_{\mathbf{X}}f_{2}(\mathbf{X})=\sum_{i=1}^{N}\sum_{j=1}^{M}X_{ij}(c_{i}+s_{ i}+b_{i}) \tag{27}\]  
We introduce weighting coefficients \(\alpha\) and \(\beta\), combining them into a single objective function:  
\[\min_{\mathbf{X}}\left(\alpha\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{j}}{b_{ i}}X_{ij}+\beta\sum_{i=1}^{N}\sum_{j=1}^{M}X_{ij}(c_{i}+s_{i}+b_{i})\right) \tag{28}\]  
We employ the Lagrangian function, as follows:  
\[L(\mathbf{X},\lambda)=\alpha\sum_{i=1}^{N}\sum_{j=1}^{M}\frac{d_{ik}^{j}}{b_{ i}}X_{ij}+\beta\sum_{i=1}^{N}\sum_{j=1}^{M}X_{ij}(c_{i}+s_{i}+b_{i})+\sum_{j=1}^{M }\lambda_{j}\left(\sum_{i=1}^{N}X_{ij}-1\right) \tag{29}\]  
The Lagrange multipliers \(\lambda_{j}\) enforce the constraint that the sum of the offloading decisions for each task \(\mathcal{T}_{j}\) equals 1. This formulation leads to the dual problem:  
\[\max_{\lambda}\min_{\mathbf{X}}L(\mathbf{X},\lambda) \tag{30}\]  
To solve this dual problem, distributed optimization techniques such as gradient descent are employed, iteratively solving for the optimal offloading decision matrix \(\mathbf{X}^{*}\) and Lagrange multipliers \(\lambda^{*}\). Given the multi-objective nature of the task offloading and resource allocation problem, we employ a distributed gradient descent algorithm enhanced with Gradient Surgery toensure convergence to a Pareto optimal solution.