## 4 ADMM-Based Distributed Solution Strategy
### Problem Decomposition and Joint Optimization Algorithm  
In this paper, the Alternating Direction Method of Multipliers (ADMM) [36] is utilized to address the interdependencies between variables in \(\mathbf{P0^{\prime}}\). This approach breaks down the problem into subproblems focused on the user side, the service node side, and the network side, solving each independently to enhance efficiency. In detail, the user side is responsible for selecting video resolution levels and choosing access points; the service node side oversees the allocation of computational resources; and the network side manages the planning of transmission paths and the allocation of power for wireless links. To construct the augmented Lagrangian function for the problem \(\mathbf{P0^{\prime}}\), we first establish independent local feasible sets \(\Pi_{z,f}\), \(\Pi_{y}\), and \(\Pi_{r,\hat{p}}\) for the variables in each sub-problem as given below:  
\[ \begin{cases}\Pi_{z,f}=\left\{\left\{z_{gm},f_{gm}^{j}\right\}|[0,1],\text{ constraints }C1,C2\right\}\\ \Pi_{y}=\left\{\left\{y_{mn}\right\}|\mathbb{R}^{+},\text{constraint }C7\right\}\\ \Pi_{r,\hat{p}}=\left\{\left\{r_{mn}^{k},\hat{p}_{l}\right\}\right|\mathbb{R}^{ +},\text{constraints }C6,C8,C9,C10\right\}\end{cases}  \tag{16}\]  
To address the coupling constraints, we introduce dual variables \(\lambda_{mm}\), \(v_{m}\), and \(\mu_{mj}\) for the relaxation of constraints C3, C4, and C5, respectively. This yields the following augmented Lagrangian function:\[\mathfrak{L}_{\rho}(\mathbf{Z},\mathbf{F},\mathbf{Y},\mathbf{R}, \mathbf{\hat{P}},\lambda,\
u,\mu) =-\mathrm{U}(\mathbf{Z})+\sum_{m\in\mathcal{M}}\sum_{n\in\mathcal{ N}}\lambda_{mn}\mathfrak{L}_{\rho_{1}}+\sum_{m\in\mathcal{M}}\
u_{m}\mathfrak{L}_{ \rho_{2}}\] \[+\sum_{m\in\mathcal{M}}\sum_{j\in\mathcal{J}}\mu_{mj}\mathfrak{L} _{\rho_{3}}+\frac{\rho_{1}}{2}\sum_{m\in\mathcal{M}}\sum_{n\in\mathcal{N}} \mathfrak{L}_{\rho_{1}}^{2}\] \[+\frac{\rho_{2}}{2}\sum_{m\in\mathcal{M}}\mathfrak{L}_{\rho_{2}}^ {2}+\frac{\rho_{3}}{2}\sum_{m\in\mathcal{M}}\sum_{j\in\mathcal{J}}\mathfrak{L }_{\rho_{3}}^{2} \tag{17}\]  
where \(\rho_{1}\), \(\rho_{2}\), and \(\rho_{3}\geq 0\) act as penalty parameters, and \(\mathfrak{L}_{\rho_{1}}\), \(\mathfrak{L}_{\rho_{2}}\), and \(\mathfrak{L}_{\rho_{3}}\) are specified by:  
\[\left\{ \begin{aligned} \mathfrak{L}_{\rho_{1}}&=\sum_{p_{mn }^{k}\in\mathcal{P}_{mn}}r_{mn}^{k}-h_{mn}y_{mn}\\ \mathfrak{L}_{\rho_{2}}&=\sum_{g=1}^{G}z_{gm}c_{g}- \sum_{n\in\mathcal{N}}y_{mn}\\ \mathfrak{L}_{\rho_{3}}&=\sum_{p_{mn}^{k}\in \mathcal{P}_{m}}a_{mj}r_{mn}^{k}-\sum_{g=1}^{G}f_{gn}^{j}v_{g}\end{aligned} \right. \tag{18}\]  
By utilizing ADMM, we can switch between optimizing the objective function and separating the problem \(\mathbf{P}\mathbf{0}^{\prime}\) into three subproblems: the video resolution and access node selection issue on the user side, the computational scheduling challenge on the service node side, and the path planning and power allocation on the network side. In particular, when tackling the user-side issue, the variables related to the service node side and network side are kept fixed, and the same approach is applied to the other subproblems. After completing the optimization, the centralized controller treats the outcomes of the distributed optimization as fixed values to address the overall problem from the controller's perspective. The algorithm ceases iterations once the stopping conditions are satisfied. This scheme is summarized in Figure 3.  
The issue of video resolution and selection of an access node can be handled separately by each user on their own. Consequently, the individual user's optimization problem is formulated as follows:  
Figure 3: Problem Decomposition. \[