Given two functions $g_1$ and $g_2$, two sets $X_1$ and $X_2$, and two matrices $A_1$ and $A_2$, ADMM solves $$\label{Eq:ProblemSolvedByADMM}
    \begin{array}{cl}
      \underset{x_1,x_2}{\text{minimize}} & g_1(x_1) + g_2(x_2) \\
      \text{subject to} & x_1 \in X_1\,\,,\,\, x_2\in X_2 \\
                        & A_1x_1 + A_2x_2 = 0\,.
    \end{array}$$ It uses the method of multipliers concatenated with an iteration of the Gauss-Seidel algorithm [@Bertsekas97-ParallelDistributed], i.e., it iterates on $k$: $$\begin{aligned}
        x_1^{k+1} &\in \arg\min_{x_1 \in X_1} L_{\rho}(x_1, x_2^{k};\lambda^k) 
                \label{Eq:ADMMAlg1}
                \\
        x_2^{k+1} &\in \arg\min_{x_2 \in X_2} L_{\rho}(x_1^{k+1}, x_2; \lambda^k)
                \label{Eq:ADMMAlg2}
                \\
        \lambda^{k+1} &= \lambda^k + \rho(A_1x_1^{k+1} + A_2x_2^{k+1})\,,
        \label{Eq:ADMMAlg3}
  \end{aligned}$$ where is the augmented Lagrangian of , $\lambda$ is the dual variable, and $\rho>0$ is a predefined parameter. More information about ADMM, including variations, applications, and related algorithms can be found, for example, in [@Bertsekas97-ParallelDistributed; @Boyd11-ADMM; @Combettes10-ProximalSplittingInSP]. The following theorem establishes its convergence.

\[Teo:ConvergenceADMM\] Assume $g_i:\mathbb{R}^{n_i}\xrightarrow{} \mathbb{R}$ is a convex function over $\mathbb{R}^{n_i}$, $X_i \subset \mathbb{R}^{n_i}$ is a closed convex set, and $A_i$ is a full column-rank matrix, for $i=1,2$. Also, assume problem  is solvable. Then, the sequence $\{(x_1^k,x_2^k,\lambda^k)\}$ generated by - converges to $(x_1^\star, x_2^\star, \lambda^\star)$, where

1.  $(x_1^\star,x_2^\star)$ solves 

2.  $\lambda^\star$ solves the dual problem of : $$\label{Eq:ADMMDualProb}
                        \underset{\lambda}{\text{minimize}}\,\,\, G_1(\lambda) + G_2(\lambda)\,,$$ where $G_i(\lambda) = \inf_{x_i 