\in X_i} (g_i(x_i) + \lambda^\top A_ix_i)$, $i=1,2$.

This theorem is more general than the versions in [@Boyd11-ADMM] and [@Bertsekas97-ParallelDistributed Prop.4.2]. Namely, [@Boyd11-ADMM] only proves objective convergence, but since it does not assume $A_1$ and $A_2$ to be full column-rank, the sequence $\{(x_1^k,x_2^k,\lambda^k)\}$ might not even converge. On the other hand, [@Bertsekas97-ParallelDistributed] proves claims $1)$ and $2)$ only for the special case when one of the $A_i$’s is the identity. Since we could not find a proof of this version of the theorem in literature, we provide it in Appendix \[App:ADMMProof\]. The proof is, however, very similar to the ones in [@Bertsekas97-ParallelDistributed; @Boyd11-ADMM].

[[**Applying ADMM.**]{}]{} Clearly, has the same format as : just make the following associations $$\label{Eq:BipAssociations}
        g_i(x_i) = \sum_{p \in \mathcal{C}_i} f_p(x_p)\,,\quad X_i = \bar{X}_i\,,\quad A_i = B_i^\top \otimes I_n\,,$$ for $i=1,2$. The following theorem guarantees that, under Assumptions \[Ass:FunctionsAndSets\]-\[Ass:Network\], problem  satisfies all the conditions of Theorem \[Teo:ConvergenceADMM\]; thus, we can use ADMM to solve .

\[Teo:BipApplyADMM\] Let Assumptions \[Ass:FunctionsAndSets\]-\[Ass:Network\] hold and consider problem . Then, for $i=1,2$, the function $\sum_{p \in \mathcal{C}_i} f_p(x_p)$ is convex over $\mathbb{R}^n$, the set $\bar{X}_i$ is closed and convex, and the matrix $B_i^\top \otimes I_n$ has full column-rank. Furthermore, problem  is solvable.

All 