I have the following optimization problem \begin{align} x^*, y^*&:= \arg\min_{\{x_i\}_i, \{y_j\}_j}~ \sum_{i \in \mathcal{N}_1} f_i(x_i) + \sum_{j \in \mathcal{N}_2} f_j(y_j)\\ \nonumber \text{s.t. } &\alpha_{i,j}(k) x_{i}(k) = \alpha_{i,j}(k) y_{j}(k), \forall k, i \in \mathcal{N}_1, m \in \mathcal{N}_2 \end{align} where $\mathcal{N}_1 \cup \mathcal{N}_2 = \{1,2,\dots,N\}$
In order to get the ADMM formulation, I wrote the lagrangian of the problem as \begin{align} \nonumber \mathcal{L}_\rho(\{x_i\}, \{y_j\},\lambda) = &\sum_{i \in \mathcal{N}_1} f_i(x_i) + \sum_{j \in \mathcal{N}_2} f_j(y_j) + \sum_k \sum_i \sum_j \lambda_{i,j}(k) \alpha_{i,j}(k)(x_{i}(k) - y_{j}(k))\\ & + \frac{\rho}{2} \sum_k \sum_i \sum_j (\alpha_{i,j}(k))^2(x_{i}(k) - y_{j}(k))^2 \end{align}
Then, I tried to write the ADMM formulation as
- For $i \in \mathcal{N}_1$: \begin{align} \nonumber x_{i}^{t+1}(k) &= \underset{x_{i}}{\arg \min}~ f_i(x_{i}) + \sum_k \sum_j \lambda_{i,j}^{t}(k) \alpha_{i,j}(k) (x_{i}(k)-y_{j}^{t}(k))\\ & + \frac{\rho}{2} \sum_k \sum_j (\alpha_{i,j}(k))^2 (x_{i}(k)-y_{j}^{t}(k))^2 \end{align}
- For $j \in \mathcal{N}_2$: \begin{align} \nonumber y_{j}^{t+1}(k) &= \underset{y_{j}}{\arg \min}~ f_j(y_{j}) + \sum_k \sum_i \lambda_{i,j}^{t}(k) \alpha_{i,j}(k) (x_{i}^{t+1}(k)-y_{j}(k))\\ & + \frac{\rho}{2} \sum_k \sum_i (\alpha_{i,j}(k))^2 (x_{i}^{t+1}(k)-y_{j}(k))^2 \end{align}
- Dual variables: \begin{align} \lambda_{i,j}^{t+1}(k) = \lambda_{i,j}^{t}(k) + \rho \alpha_{i,j}(k) (x_{i}^{t+1}(k) - y_{j}^{t+1}(k)) \end{align}
I want to check if the formulation I'm making is valid or not. Thanks!