\bar{X}_c\,,\quad c = 1,\ldots,C \\
                                  & \sum_{c=1}^C (B_c^\top \otimes I_n)\bar{x}_c= 0\,,
            \end{array}$$ where the variable is $(\bar{x}_1,\ldots,\bar{x}_C) \in (\mathbb{R}^n)^P$. To solve , we cannot apply ADMM - directly, but its generalization to solve $$\begin{array}{cl}
        \underset{x_1,\ldots,x_C}{\text{minimize}} & g_1(x_1) + \cdots + g_C(x_C) \\
        \text{subject to} & x_c \in X_c\,,\quad c=1,\ldots,C \\
                          & A_1x_1 + \cdots + A_Cx_C = 0\,.
      \end{array}$$ Such a generalization is straightforward, but there are no convergence guarantees in the sense that there is no known proof of Theorem \[Teo:ConvergenceADMM\] for the resulting algorithm. Yet, experimental results, as the ones presented here, suggest that Theorem \[Teo:ConvergenceADMM\] might still hold.

The following algorithm is a straightforward generalization of Algorithm \[Alg:ADMMBipartite\] to solve .

\[Alg:DADMMGeneral\]

for all $p \in \mathcal{V}$, set $\gamma_{p}^{1} = x_p^{1} = 0$ and $k=1$ $$v_p^{k} = \gamma_p^{k}-
                \rho \sum_{\begin{subarray}{c}
                             j \in \mathcal{N}_p \\
                             j < p
                           \end{subarray}
                }x_j^{k+1} - \rho \sum_{\begin{subarray}{c}
                             j \in \mathcal{N}_p \\
                             j > p
                           \end{subarray}
                }x_j^{k}$$ \[SubAlg:GeneralDADMMv\] and find $$x_p^{k+1} = \begin{array}[t]{cl}
                            \textrm{argmin} & f_p(x_p) + {v_p^{k}}^\top x_p + \frac{D_p \rho}{2}\|x_p\|^2\\
                            \textrm{s.t.} & x_p \in X_p
                          \end{array}$$ \[SubAlg:ADMMGenProb\] Send $x_p^{k+1}$ to $\mathcal{N}_p$ \[SubAlg:ADMMGenComm\]

$
            \gamma_p^{k+1} = \gamma_p^{k} + \rho \sum_{j \in \mathcal{N}_p} (x_p^{k+1} -  x_j^{k+1})
        $ \[SubAlg:ADMMGenDualVarUp\] $k \gets k+1$

In Algorithm 