## 6 E-DDTMD Algorithm Based on ADMM
### E-Ddtmd Algorithm Based on ADMM
Therefore, for SN i, the following equivalent optimization problem is required to be solved when updating the local variables at iteration \((t+1)\):  
\[\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}\left\{ v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\right.\] \[+\sum\limits_{k\in\mathcal{I},j\in\mathcal{J}}\delta^{i}_{k,j}( \hat{b}^{i}_{k,j}-b_{k,j})+\sum\limits_{k\in\mathcal{I},j\in\mathcal{J}}\sigma^ {i}_{k,j}(\hat{d}^{i}_{k,j}-d_{k,j}) \tag{35a}\] \[+\frac{\rho}{2}\sum_{k\in\mathcal{I},j\in\mathcal{J}}\left\|\hat{b}^{i}_{k,j}-b _{k,j}\right\|^{2}+\frac{\rho}{2}\sum_{k\in\mathcal{I},j\in\mathcal{J}}\|\hat{d} ^{i}_{k,j}-d_{k,j}\|^{2}\},\] \[S.t.\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}\in\varphi_ {i}. \tag{35b}\]  
Obviously, problem (35) is a convex problem, so the optimal solution can be obtained by using the primal-dual interior-point algorithm or the CVX toolkit [33]. (2) Update the \(\left\{\mathbf{b},\mathbf{d}\right\}\) global variables:  
In this step, we need to update the global variables. Through the previous steps, we get the local variable \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}^{[t+ 1]}\); for a given \(\left\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\right\}_{i\in\mathcal{I}}^{[t+ 1]}\), a global variable \(\left\{\mathbf{b},\mathbf{d}\right\}\) updated based on the following formula:  
\[\left\{\mathbf{b}\right\}^{[t+1]}=\underset{\{b_{k,j}\}}{\arg}\underset{\{b_{k,j} \}}{\min}[\sum_{i\in\mathcal{I}}\sum\sum_{k\in\mathcal{I},j\in\mathcal{J}} \delta^{i[t]}_{k,j}(\hat{b}^{[t+1]}_{k,j}-b_{k,j})+\frac{\rho}{2}\sum_{i\in \mathcal{I}}\sum_{k\in\mathcal{I},j\in\mathcal{J}}\|\hat{b}^{i[t+1]}_{k,j}-b_{ k,j}\|^{2}]. \tag{36}\]Since the above is an unconstrained quadratic convex problem, we can solve it by simply setting the gradient of \(\{\mathbf{b}\}\) and \(\{\mathbf{d}\}\) to zero, i.e.,  
\[\sum_{i\in\mathcal{I}}\delta_{k,j}^{[t]}+\rho\sum_{i\in\mathcal{I}}(\hat{b}_{k,j }^{[t+1]}-b_{k,j})=0,\forall k,j, \tag{38}\]  
\[\sum_{i\in\mathcal{I}}\sigma_{k,j}^{[t]}+\rho\sum_{i\in\mathcal{I}}(d_{k,j}^{[ t+1]}-d_{k,j})=0,\forall k,j, \tag{39}\]  
This leads to  
\[b_{k,j}^{[t+1]}=\frac{1}{I\rho}\sum_{i\in\mathcal{I}}\delta_{k,j}^{[t]}+\frac{ 1}{I}\sum_{i\in\mathcal{I}}\hat{b}_{k,j}^{[t+1]},\forall k,j, \tag{40}\]  
\[d_{k,j}^{[t+1]}=\frac{1}{I\rho}\sum_{i\in\mathcal{I}}\sigma_{k,j}^{[t]}+\frac{ 1}{I}\sum_{i\in\mathcal{I}}\tilde{d}_{k,j}^{[t+1]},\forall k,j. \tag{41}\]  
By the Lagrange multiplier in the first \(t\) iteration initialized to zero, namely, the \(\sum_{i\in\mathcal{I}}\delta_{k,j}^{[t]}=0\) and \(\sum_{i\in\mathcal{I}}\sigma_{k,j}^{[t]}=0,\forall k,j\), we can make the type reduction for:  
\[\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}\left\{ v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\right.\]0  
\[\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}\left\{ v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\right.\]1
3. The Lagrange multiplier \(\left\{\mathbf{\delta}^{i},\mathbf{\sigma}^{t}\right\}_{i\in\mathcal{I}}\) is updated: The Lagrange multiplier is updated according to the following formula: \[\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}\left\{ v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\right.\]2 (44) \[\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}\left\{ v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\right.\]3 (45)
4. Iteration stopping criterion: The original residual of each SN must be as small as possible under feasible conditions. Therefore, we set the iteration stopping threshold as \[\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}\left\{ v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\right.\]4 (46) \[\underset{\{a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}}\}}{\min}\left\{ v_{i}(a_{i},\mathbf{\hat{b}^{i}},\mathbf{\hat{d}^{i}})\right.\]5 (47)  
where \(\varepsilon_{pri}>0\) is the threshold in primal feasibility conditions.