## 2 Background
### Alternating Direction Method of Multiplier (ADMM)  
The algorithm was introduced in 1970 by [PERSON], [PERSON], [PERSON], and [PERSON] [21]. ADMM is a simple algorithm that incorporates the features of Dual Ascent and the Method of Multipliers. The powerful feature of dual ascent is that in some cases, decomposition can be applied. The method of multipliers, on the other hand, provide robustness to the the dual ascent in a sense that convergence can be achieved without strict assumptions on the objective function. Therefore, ADMM can be used towards the solution of a multi-area state estimation, as the interconnected system measurement model in a distributed architecture. The general form of ADMM can be written as follows [21]:  
\[\begin{split} minimize& f(x)+g(z)\\ \text{subject to}& Ax+Bz=c\end{split} \tag{12}\]  
with variables \(x\in\mathbb{R}^{n}\) and \(z\in\mathbb{R}^{m}\), where \(A\in\mathbb{R}^{p\times n}\), \(B\in\mathbb{R}^{p\times m}\), and \(c\in\mathbb{R}^{p}\). By using the method of multiplier, the augmented Lagrangian can be written as follows:  
\[L_{\rho}(x,z,v):=f(x)+g(z)+v^{T}(Ax+Bz-c)+\left(\frac{\rho}{2}\right)\left\| Ax+Bz-c\right\|_{2}^{2}. \tag{13}\]  
Therefore, the structure of ADMM consists of the following iterations:  
\[x^{t+1}:=\operatorname*{arg\,min}_{x}L_{\rho}\Big{(}x,z^{t},v^{t}\Big{)} \tag{14}\]  
\[z^{t+1}:=\operatorname*{arg\,min}_{z}L_{\rho}\Big{(}x^{t+1},z,v^{t}\Big{)} \tag{15}\]  
\[v^{t+1}:=v^{t}+\rho\Big{(}Ax^{t+1}+Bz^{t+1}-c\Big{)} \tag{16}\]  
where \(\rho>0\). The iteration in (14) is just a minimization over \(x\) while fixing other variables, i.e., \(z\) and \(v\). Then, a minimization over \(z\) is performed in (15) while fixing \(v\) and using updated \(x\) from (14). The last iteration is simply an update of the dual variable \(v\). Actually, the dual variable can be viewed as a running sum of the error, analogous to integrator controller in control theory. This version of ADMM is often called unscaled form. Another version of ADMM known as scaled version can be obtained. The process starts by defining the residual \(r=Ax+Bz-c\), and combining the linear and quadratic terms in the augmented Lagrangian function to be as follows:  
\[\begin{split} v^{T}r+\left(\frac{\rho}{2}\right)\left\|r\right\| _{2}^{2}&=\left(\frac{\rho}{2}\right)\left\|r+\left(\frac{1}{ \rho}\right)v\right\|_{2}^{2}-\left(\frac{1}{2\rho}\right)\left\|v\right\|_{ 2}^{2}\\ &=\left(\frac{\rho}{2}\right)\left\|r+u\right\|_{2}^{2}-\left( \frac{\rho}{2}\right)\left\|u\right\|_{2}^{2}\end{split} \tag{17}\]  
where \(u=\left(\frac{1}{\rho}\right)v\) is the scaled dual variable. Hence, the ADMM cycles in the scaled version can be expressed as follows:  
\[x^{t+1}:=\operatorname*{arg\,min}_{x}\left(f(x)+\frac{\rho}{2}\left\|Ax+Bz^{ t}-c+u^{t}\right\|_{2}^{2}\right) \tag{18}\]  
\[z^{t+1}:=\operatorname*{arg\,min}_{z}\left(g(z)+\frac{\rho}{2}\left\|Ax^{t+1}+ Bz-c+u^{t}\right\|_{2}^{2}\right) \tag{19}\]  
\[u^{t+1}:=u^{t}+\left(Ax^{t+1}+Bz^{t+1}-c\right) \tag{20}\]  
From the provided two versions of ADMM iterations, both approaches are equivalent. The scaled form, however, is relatively shorter regarding formulation size.