in any connected network).

We will see that D-ADMM requires almost always less communication steps to achieve a given accuracy than Algorithm \[Alg:Giannakis\], for several instances of .

Another ADMM-based algorithm is [@Schizas08-ConsensusAdHocWSNsPartI] (also appearing in [@Schizas08-ConsensusAdHocWSNsPartII; @Erseghe11-FastConsensusByADMM]). That algorithm, in contrast with D-ADMM and Algorithm \[Alg:Giannakis\], uses two communication steps per iteration. Since the number of iterations it takes to achieve a given accuracy in the average consensus problem is similar to Algorithm \[Alg:Giannakis\] [@Erseghe11-FastConsensusByADMM], it takes the double of the communication steps.

[[**Double-looped algorithms.**]{}]{} Several double-looped algorithms were proposed to solve particular optimization problems, which as D-ADMM, can be generalized to solve . In particular, they can solve , by solving the following dual problem of : $$\label{Eq:DualProblemCloned}
    \underset{\lambda}{\text{maximize}} \,\,\, L_{\rho}(\lambda)\,,$$ where $L_\rho(\lambda)$ is the (augmented) dual function $$\label{Eq:LagrangianProblemCloned}
    L_\rho(\lambda) = 
    \begin{array}[t]{cl}
      \underset{\bar{x}}{\inf} & \sum_{p=1}^P f_p(x_p) + \sum_{\{i,j\}\in \mathcal{E}} \phi_{\lambda_{\{i,j\}}}^\rho (x_i - x_j) \\
      \text{s.t.} & x_p \in X_p\,,\quad p=1,\ldots,P\,,
    \end{array}$$ and $\phi_{v}^\rho(\eta) := v^\top \eta + \frac{\rho}{2}\|\eta\|^2$. It can be proved that $L_\rho(\lambda)$ is differentiable, and thus  can be solved with a gradient method, or even with a fast (Nesterov) gradient method. The outer loop consists of applying such algorithm: it updates the dual variables $\lambda_{\{i,j\}}$. However, 