## 4 ADMM-Based Distributed Solution Strategy
### Problem Decomposition and Joint Optimization Algorithm
* Service node-side optimization: Service nodes distribute computational resources to users. * Network-side optimization: The network decides on transmission paths and administers power for the wireless links. Once these subproblems are solved, the dual variables are updated, and the cycle is repeated until the system converges. The subsequent algorithm outlines the ADMM-based distributed approach for managing resource distribution and coordinating interference within the GEO-LEO satellite system. After solving the problem, the user's video resolution level \(z_{gm}\) and the new variable \(f_{gm}^{j}\) are converted back to binary variables based on the marginal benefits of the obtained linear solution. Here, considering that the new variable \(f_{gm}^{j}\) represents the selection of access base station \(j\) for the video file requested by user \(m\) at level \(g\), it is necessary to first determine the video quality level \(g\) chosen by user \(m\) based on \(z_{gm}\) before obtaining the access node selection variable \(x_{mj}\). Then, find the submatrix corresponding to \(\left\{f_{gm}^{j}\right\}\) based on parameters \(m\) and \(g\), which is the set corresponding to the access node selection variable \(x_{mj}\). The complete algorithm flow is shown in Algorithm 1. ```
1:Input: Network \((\mathcal{D},\mathcal{L})\), user video demand \(f\). 2:Initialization Initialize variables \(\mathbf{Z}(0)\), \(\mathbf{X}(0)\), \(\mathbf{Y}(0)\), \(\mathbf{R}(0)\), and \(\hat{\mathbf{P}}(0)\), as well as dual variables \(\lambda_{mn}(0)\), \(\upsilon_{m}(0)\), and \(\mu_{mj}(0)\). Set stopping threshold \(\epsilon\) and maximum iterations \(T\). 3:for each step \(t\leq T\)do
4: Broadcast dual variables \(\lambda_{mn}(t)\), \(\upsilon_{m}(t)\), and \(\mu_{mj}(t)\) from the SDN controller to users and service nodes. 5: Solve the user-side optimization problem **P1** for each user to obtain updated video resolution \(\mathbf{Z}(t+1)\) and access point selection \(\mathbf{F}(t+1)\). 6: Solve the service node-side optimization problem **P2** to obtain updated computational resource allocation \(\mathbf{Y}(t+1)\). 7: Solve the network-side optimization problem **P3** to update transmission paths \(\mathbf{R}(t+1)\) and power allocation \(\hat{\mathbf{P}}(t+1)\). 8: Update the dual variables \(\lambda_{mn}(t+1)\), \(\upsilon_{m}(t+1)\), and \(\mu_{mj}(t+1)\) using the results from step 4. 9:if the difference in the objective function value between iterations \(t\) and \(t-1\) is less than \(\epsilon\)then
10: stop
11:endif
12:endfor
13:Output: Video resolution selection \(\mathbf{Z}\), access point selection \(\mathbf{X}\), resource allocation \(\mathbf{Y}\), transmission paths \(\mathbf{R}\), and power allocation \(\hat{\mathbf{P}}\). ```  
**Algorithm 1** Algorithm for ADMM-based Resource Allocation and Interference CoordinationTo make the proposed scheme more accessible, consider a simplified dual-layer satellite scenario with two users (\(U1\) and \(U2\)), one LEO satellite acting as a service node with limited computational and caching resources, one GEO satellite functioning purely as a communication relay, and one ground base station connected to a small data center. We assume:  
* Each user requests video service at one of two resolution levels (e.g., HD or SD). * The LEO satellite and data center can process and cache video content, while the GEO satellite primarily forwards traffic. * Spectrum resources must be shared among all links, and interference arises if both satellites transmit in the same frequency band simultaneously. We can formulate a smaller version of our optimization model for this scenario as follows:  
* Decision variables: \(xU1,xU2\in 0,1\) to indicate whether each user is assigned to SLEO or BS for primary video access. \(rU1,rU2\) to represent the chosen resolution levels (e.g., 1 for SD, 2 for HD). * Objective: Maximize overall video quality while minimizing interference and respecting power/bandwidth constraints at SLEO and BS.