## 4 ADMM-Based Distributed Solution Strategy
### Algorithm Performance Analysis  
The problem \(\mathbf{P0^{\prime}}\) is a convex optimization problem featuring both a convex objective function and constraints, thus ensuring convergence to a globally optimal solution. Moreover, the augmented Lagrangian approach supports strong duality, meaning that the ADMM-based algorithm achieves convergence to the global optimum.  
Regarding complexity analysis, utilizing centralized methods like primal-dual interior-point algorithms to solve \(\mathbf{P0^{\prime}}\) would entail a complexity of \(\mathcal{O}\big{(}(MNKL)^{3}\big{)}T_{1}\), where \(T_{1}\) denotes the number of iterations, \(M\) is the user count, \(N\) the number of service nodes, \(K\) the number of candidate paths, and \(L\) represents the number of links. However, by applying ADMM for problem decomposition, complexity is notably diminished, allowing for distributed solving at the level of each component (user, service node, and network).  
The complexity for resolving the user-side sub-problem \(\mathbf{P1}\) via a convex optimization tool is \(\mathcal{O}\big{(}(MGJ)^{3}\big{)}\), where \(G\) stands for the number of video quality levels, and \(J\) the number of access points. The service node-side sub-problem \(\mathbf{P2}\) has a complexity of \(\mathcal{O}\big{(}(MNN)^{3}\big{)}\). Lastly, solving the network-side sub-problem \(\mathbf{P3}\) carries a complexity of \(\mathcal{O}\big{(}(MNKL)^{3}\big{)}T_{1}\).  
Consequently, the overall complexity of each iteration of the proposed ADMM-based algorithm is \(\mathcal{O}\big{(}(MNKL)^{3}\big{)}T_{1}\). With \(T_{2}\) iterations for achieving convergence, the total complexity becomes \(\mathcal{O}\big{(}(MNKL)^{3}\big{)}T_{1}T_{2}\). This highlights that the ADMM algorithm can substantially reduce computational complexity while ensuring effective resource allocation and interference coordination in a GEO-LEO satellite network.  
In addition to providing a detailed complexity analysis of our ADMM-based algorithm, we compared its computational efficiency with two commonly referenced methods in the literature.  
Centralized Interior-Point Method. Classic interior-point solvers applied to similar optimization problems demonstrate a polynomial time complexity that grows significantly with the problem size \(\mathcal{O}\big{(}(MNKL)^{3}\big{)}\) per iteration. While these solvers can converge in relatively few iterations, they rely on a centralized structure, leading to high memory overhead and potentially long solution times for large-scale heterogeneous satellite networks.  
Gradient-Based Distributed Approach. Distributed gradient descent or primal-dual methods operate without forming large Hessian matrices, reducing per-iteration costs. However, these methods often need more iterations to achieve convergence and may suffer from slow progress in the presence of highly coupled constraints, as found in dual-layer satellite networks.  
By decomposing the resource allocation and interference coordination problem into subproblems (user side, service node side, and network side), the ADMM-based algorithm strikes a balance between per-iteration complexity and convergence speed. Consequently, the total complexity per iteration is lower than that of the centralized interior-point approach, while convergence is typically faster than standard gradient-based methods in scenarios with tightly coupled constraints. This efficiency is particularly advantageous for large-scale GEO-LEO satellite networks.