While  [68] studied a general multi-block coupled problem (<ref>) in nonconvex setting, some other  works considered some special cases. For example,  [113] studied the convergence of classical ADMM for a special two-block nonconvex optimization without any coupled objective, i.e., $n=1$, $g=0$, and $\B=\mathbf{I}$.
Besides, [114] studied a special $3$-block nonconvex optimization  arising from image processing with $n=3$,  $g=0$ and $h$ strongly convex and quadratic. These works can be regarded as the special cases of [68]  in terms of both problem structures and convergence conditions.

§.§.§ Flexible ADMM
Flexible ADMM  can be viewed as a flexible version of classical ADMM  [115]. The key idea is to activate the computing agents by a random mechanism.
This is often preferred when the computing agents have limited energy or suffer  certain random ON/OFF behaviors. The implementation of
flexible ADMM is presented below.

\begin{align*}
&\textbf{Flexible ADMM:} ~~\\
& \textbf{Primal update:}~ \\
& \x_i^{k+1} \!\!=\!\!
\begin{cases}
\arg \min_{\x_i}\!\Lag_{\rho}(\x_{<i}^{k+1}, \! \x_i, \! \x_{>i}^k, \!\y^k, &\!\!\!\!\!\boldlambda^k), \text{ if agent  $\x_i$ is activated.} \\
\x_i^k,  &~~~\text{ if agent $\x_i$ not activated.}
\end{cases}\\
& \y^{k+1} \!\!=\!\! \begin{cases}
\arg \min_{\y} \Lag_{\rho}(\x^{k+1}, \y, \boldlambda^k),  &\quad \quad  \text{ if agent $\y$ is activated.} \\
\y^k, & \quad \quad  \text{ if agent  $\y$  not activated.}
\end{cases}\\
&\textbf{Dual update:}~   \boldlambda^{k+1} = \boldlambda^k + \rho (\A \x^{k+1} + \B \y^{k+1})
\end{align*}

Flexible ADMM employs a random agent activation mechanism in the primal update. As a result, a decision block will be updated only if the related computing agent is activated, otherwise the old update will be held and communicated to the others at each iteration.  Clearly, the convergence of flexible ADMM depends on the random agent activation mechanism. For the two usual cases that the agents are  all  activated  with  positive probability   or  in an essentially cyclic manner  (i.e., activated at least once within $K$ iterations),
the convergence of flexible ADMM towards stationary points has been  established for a class of problems with the following structures [115].

a)  $f_i$ are convex or $\Lips$ differentiable.

b)  $h$ is $\Lips$ differentiable and $g=0$.

c) $\A_i$ have full column rank.

d) $\B = \mathbf{I}$.

Note that the above conditions a)-d) are a special case of conditions a)-d) with classical ADMM.  To our understanding, the more general conditions  a)-d) with classical ADMM are readily extensible to the flexible ADMM since the two methods share the same procedures  to establish convergence and the related conditions  play the same roles.

§.§.§ Bregman ADMM
For multi-block coupled problem (<ref>),  Bregman ADMM is another  ADMM variant for nonconvex optimization  [116]. Central to the method is to regularize the subproblems with classical ADMM by some Bregman divergence. Bregman divergence or Bregman distance is one metric for measuring the discrepancy of two points, which includes Eculidean distance as a special case.  Compared with classical ADMM,  Bregman ADMM does not impose any restrictions on the  objective functions $f_i$ (i.e., restricted prox-regular) and the coefficient matrices $\A_i$, making the method  applicable to more general problems. Bregman ADMM takes the following general framework.

\begin{align*}
&\textbf{Bregman ADMM:} ~~\\
&\textbf{Primal update:}~  \\
&~\x_i^{k+1} = \arg \min\limits_{\x_i}\Lag_{\rho}(\x_{<i}^{k+1}, \x_i, \x_{>i}^k, \y^k, \boldlambda^k) + \Delta_{\phi_i}(\x_i, \x_i^k), \forall i \in N.\\
&~\y^{k+1} = \arg \min\limits_{\y} \Lag_{\rho}(\x^{k+1}, \y, \boldlambda^k) + \Delta_{\psi}(\y, \y^k)\\
& \textbf{Dual update:}~   \boldlambda^{k+1} = \boldlambda^k + \rho (\A \x^{k+1} + \B \y^{k+1})
\end{align*}

where $\Delta_{\phi_i}(\cdot, \cdot)$ and  $\Delta_{\psi}(\cdot, \cdot)$ are  Bregman divergence   defined on the  convex functions $\phi_i$ and $\psi$, respectively.  For a general  convex differentiable function $\phi:\R^n \rightarrow \R$,  the related  Bregman divergence is defined by

\begin{align*}
\Delta_{\phi}(\x, \y) = \phi(\x) - \phi(\y) - \langle \nabla \phi(\y), \x - \y\rangle, ~\forall \x, \y \in \R^n.
\end{align*}

Note that  Bregman distance reduces to Eculidean distance  $\Delta_{\phi}(\x, \y) = \Vert \x - \y\Vert^2$ with $\phi(\x) = \Vert \x \Vert^2$,  which has been widely used in proximal ADMM variants as discussed before. For  $\phi(\x) = \Vert \x \Vert^2_{\Q}$, we have the resulting Bregman distance $\Delta_{\phi}(\x, \y) = \Vert \x - \y \Vert^2_{\Q}$  which has been used in many generalized ADMM  variants.  From this perspective, we see that the Bregman ADMM defines a  general  algorithmic  framework for solving problem (<ref>). The practical implementation requires to determine the specific Bregman divergence  $\Delta_{\phi_i}(\cdot, \cdot)$ and  $\Delta_{\psi}(\cdot, \cdot)$.
The convergence of  Bregman ADMM towards first-order stationary points has been established for a class of problems with the following structures [116].

a)  $g=0$.

b)  $h$, $\phi_i$, $\varphi$ are $\Lips$ differentiable.

c)  $f_i$ or $\phi_i$,  $h$ or $\varphi$ are strongly convex.

d)  $\B$ has full row rank .

Consider a usual  case that Eculidean distance is used as  Bregman divergence (i.e., $\phi_i(\x) = \Vert \x \Vert^2$ and $\varphi(\x) = \Vert \x \Vert^2$), we clearly have $\phi_i$ and $\varphi$ strongly convex and $\Lips$ differentiable. In such context, conditions  a)-d) reduce to $g=0$, $h$ $\Lips$ differentiable and $\B$ has full row rank.  Actually, this represents a  special case of conditions b)-d) with classical ADMM.  To our understanding,  the general conditions  b)-d) with classical ADMM can be directly extended to Bregman ADMM if some strongly convex and $\Lips$ differentiable Bregman divergence functions  $\phi_i$ and $\varphi$ are selected. This is because the two methods follow the same procedures  to establish convergence and the related conditions play similar roles.
One benefit resulting from Bregman divergence  is that the restricted prox-regular conditions on $f_i$ with classical ADMM and the rank assumptions on the coefficient matrices $\A_i$ can be relaxed, making the methods applicable to more general problems.

The above results are for the general  Bregman ADMM. Some works have studied
some special instances of Bregman ADMM.  For example,  [117] proposed two proximal ADMM variants where  the proximal terms $\frac{\tau}{2}\Vert \x_i - \x_i^k \Vert^2_{\PP_i}$ and $\frac{L_{\y}}{2}\Vert \y - \y^k\Vert^2$ are used as Bregman divergence. The convergence of the methods was proved under the conditions: i) $g$ and $h$ are $\Lips$ differentiable, and ii) $\B$ has full row rank, which are in line with a) - d).

§.§.§ Proximal linearized ADMM
Note that the above ADMM variants for  problem (<ref>) require to solve the subproblems exactly at each iteration. This could be expensive or not efficient when the objective functions are complex and coupled.  To  reduce per-iteration complexity,  [79] proposed a proximal linearized ADMM as an inexact ADMM variant. The key idea is to linearize some complex (i.e., often hard to compute proximal mappings) but differentiable parts of the objective components and then solving the resulting easier proximal linearized subproblems.  Proximal linearized ADMM takes the following iterative scheme.

\begin{align*}
&\textbf{Proximal linearized ADMM:} ~~\\
&\textbf{Primal update:}~ \\
&  \x_i^{k+1} \!=\! \arg \min_{\x_i}
f_i(\x_i) \!+\! l_i(\x^k, \y^k, \boldlambda^k)^\top(\x_i \!-\!\x_i^k) \!+\! \frac{\tau_{\x}}{2}\Vert \x_i \!-\!\x_i^k \Vert^2, i \in N.\\
&  \y^{k+1} \!=\! \arg \min_{\y} u(\x^{k + 1}, \y^k, \boldlambda^k)^\top(\y- \y^k) + \frac{\tau_{\y}}{2} \Vert \y - \y^k \Vert^2 \\
& \textbf{Dual update:}~   \boldlambda^{k+1} = \boldlambda^k + \rho ( \A \x^{k+1} + \B \y^{k+1})
\end{align*}

where  $l_i(\x^k, \y^k, \boldlambda^k)$ and $ u(\x^{k+1}, \y^k, \boldlambda^k)$
denote the gradients of the differentiable parts of AL function w.r.t. $\x_i$ and $\y$ respectively. Specifically,  $l_i(\x^k, \y^k, \boldlambda^k)$ is obtained by differentiating $g(\x, \y) + (\boldlambda^k)^\top \A \x + \frac{\rho}{2}\Vert\A \x + \B \y \Vert^2$ w.r.t. $\x_i$ at $(\x^k, \y^k)$,  and $u(\x^{k+1}, \y^k, \boldlambda^k)$ is obtained by differentiating $h(\y) + g(\x, \y) + (\boldlambda^k)^\top \B \y + \frac{\rho}{2}\Vert \A \x + \B \y \Vert^2$ w.r.t. $\y$ at $(\x^{k+1}, \y^k)$. Here we consider a general  $f_i$ that is probably nonsmooth. If $f_i$ is smooth and differentiable, it can be linearized  and involved in $l_i(\x^k, \y^k, \boldlambda^k)$ similarly.
Since a first-order linearization only provides an inaccurate approximation to the AL function, some proximal terms $\frac{\tau_{\x}}{2}\Vert \x_i - \x_i^k \Vert^2$ and $\frac{\tau_y}{2}\Vert \y - \y^k\Vert^2$ are required to control the approximation accuracy.

Slightly different from classical ADMM where (<ref>) is considered as a multi-block optimization and $\gauss$ decomposition is used to enable a multi-block serial update, proximal linearized ADMM was derived by  treating (<ref>) as a two-block optimization and applying $\gauss$ decomposition to the two decision blocks  $\x:=(\x_i)_{i=1}^n$ and $\y$. Due to the linearization, we have subproblems  $\x$  naturally decomposable across the decision components $\x_i$. This brings the benefit of parallel implementation.   Note that  subproblems $\x_i$ in the primal update reduce to  the proximal mappings of $f_i$, i.e., $\x_i^{k+1} = \arg \min_{\x_i} f_i(\x_i) + \frac{\tau_x}{2}\Vert \x_i - (\x_i^k - \tau_{\x}^{-1} l_i(\x^k, \y^k, \boldlambda^k)) \Vert^2$, and subproblems $\y$ reduce to a gradient-like iterate, i.e., $\y^{k+1} = \y^k - \tau_{\y}^{-1} u(\x^{k+1}, \y^k, \boldlambda^k)$. If $f_i$ has easily computable proximal mappings, we would have  low per-iteration complexity with the method.  The convergence of proximal linearized ADMM towards stationary points was proved under the conditions [79]:

a)$g$ and $h$ are $\Lips$ differentiable.

b) ${\rm Im}(\A) \subseteq {\rm Im}(\B)$, and $\B$ has full column rank.

Similar to other linearized ADMM variants discussed before, the linearization  technique with the method also requires  the objective functions to be $\Lips$ differentiable.

One salient feature of proximal linearized ADMM is that no restrictions on the objective functions $f_i$ and the coefficient matrices $\A_i$ are required  as with classical ADMM.
This is caused by the proximal regularization  $\frac{\tau_{\x}}{2}\Vert \x - \x^k\Vert^2$ that enhances the convexity of subproblems.  As discussed with classical ADMM,
the full column rank  $\B$ can be replaced by the weaker assumption, i.e., the mapping $H(\uu) = \{\arg \min_{\y} \phi(\x, \y): \B \y = \uu\}$ is unique and $\Lips$ smooth. This holds for proximal linearized ADMM since the two methods follow  the same procedures to establish convergence and the related assumptions play the same roles.
This implies that proximal linearized ADMM depends on weaker conditions than classical ADMM.  As a result, the method is applicable to more general applications, such as sparsity regression [118] and integer programming [79]. Due to the low per-iteration complexity and parallel implementation, proximal linearized ADMM was shown  to be advantageous than  classical ADMM  in computation efficiency [79].

ADMM variants for solving multi-block coupled problem (<ref>)

Methods      Main assumptions      Types     Convergence     Features       References

[l]Majorized ADMM       [l] $f, g$ and $h$ convex.
$g$ $\Lips$ differentiable.       [l]$\Jacobian$
or Gauss-Seidel     [l]Global convergence.
Global optima.
Convergence rate $\mathcal{O}(1/k)$.       [l]Convex.    [112, 111]

Classic ADMM       [l]$f_i$ restricted prox-regular.
$g$ and $h$  $\Lips$ differentiable.
$\A_i$ full column rank
or $F_i(\uu)$ unique and $\Lips$ smooth.
$\text{Im}(\A) \subseteq \text{Im}(\B)$.
$\B$ full column rank
or $H(\uu)$ unique and $\Lips$ smooth.        Gauss-Seidel     [l]Global convergence.
Stationary points     [l] Weak assumptions.
Nonconvex.     [68]

[l]$g =0, n =1$.
h $\Lips$ differentiable.
$\A_1$ full column rank.
$\B = \mathbf{I}$       Gauss-Seidel     [l]Global convergence.
Stationary points     [l]2-block.
No restrictions on $f$.
Nonconvex.        [113]

[l]$g =0, n =3$.
$h$ quadratic .
$f_1$ convex.
$f_2$ nonconvex and nonsmooth.
$\A_1$ and $\A_3$ full row rank.      Gauss-Seidel     [l]Global convergence.
Stationary ponts     [l]3-block.
Nonconvex.        [114]

Flexible ADMM       [l]g = 0. $\B = \mathbf{I}$.
$f_i$ convex or $\Lips$ differentiable.
$\A_i$ has full column rank.       Gauss-Seidel     [l]Global convergence.
Stationary points.       [l]
Random agent activation.
Nonconvex.     [115]

2*[l]Bregman ADMM       [l]g = 0.
$f_i$ or $\phi_i$, $h$ or $\psi$ strongly convex.

$h$, $\phi_i$, $\psi$ $\Lips$ differentiable.

$\B$ has full row rank.       Gauss-Seidel     [l]Global convergence.
Stationary points.     [l]No restrictions on $f_i$ and $\A_i$.
Nonconvex.     [116]

[l] $g$ and $h$  $\Lips$ differentiable.

$\B = \mathbf{I}$       Gauss-Seidel     [l]Global convergence.
Stationary points.     [l]No restrictions on $f_i$ and $\A_i$.
Nonconvex.     [117]

linearized ADMM       [l] $g$ and $h$  $\Lips$ differentiable.
$\text{Im}(\A) \subseteq \text{Im}(\B)$.
$\B$  full column rank.        [l]Gauss-Seidel
+ Jacobian     [l]Global convergence.
Stationary points.       [l]No restrictions on $f_i$ and $\A_i$.
Low per-iteration complexity.
Nonconvex.    [79]

Summary: This section surveys ADMM variants  for solving multi-block coupled problem (<ref>). Several ADMM variants have been developed either for   convex or nonconvex  optimization. We report those ADMM variants in terms of  main assumptions, decomposition schemes (i.e., type), convergence properties, main features and references in TABLE <ref>. From the results, we draw the following overview.
For convex optimization, majorized ADMM is the main solution and guarantees the convergence towards global optima. For nonconvex optimization, classical ADMM, flexible ADMM, Bregman ADMM and proximal linearized ADMM are applicable and ensure the convergence towards stationary points.
The nonconvex ADMM variants can be distinguished by their different features. Specifically, Bregman ADMM and proximal linearized ADMM do not require any restrictions on the objective functions $f_i$ and the coefficient matrices $\A_i$ and thus are applicable to broader applications. Proximal linearized ADMM is favorable for its low per-iteration complexity. Flexible ADMM enables random agent activation mechanism which is beneficial when the computing agents have limited energy or suffer certain random ON/OFF behaviors.
Particularly, the ADMM variants for nonconvex optimization  generally depend on the two necessary conditions to ensure convergence: i) the last block $\y$ is unconstrained and $\Lips$ differentiable, and ii) ${\rm Im}(\A)\subseteq {\rm Im}(\B)$, $\B$ has full column rank or the mapping $H(\uu) = \{\arg \min_{\y} \phi(\x, \y): \B \y = \uu\}$ is unique and $\Lips$ smooth.

§.§ Consensus optimization

Previously, we have focused on multi-block optimization where the  agents hold  disjoint decision variables. In practice, it is also common to see multi-agent systems where the agents share commonly-owned decision variables but hold private objectives.  The leads to the well-known consensus optimization that takes the general formulation of

\begin{align}
\label{pp:p4} ~& \min_{\x}  F(\x) = \sum_{i = 1}^n f_i(\x) \tag{$\PP4$}
%&{\rm s.t.}~ \x \in \X_1 \cap \X_2 \cdots \cap \X_N. \notag
\end{align}

where $f_i: \R^n \rightarrow \R$ denote the private objectives of the agents;  $\x \in \R^n$ is the commonly-owned decision variables;  $f_i$ may be nonsmooth and a usual case is that some local constraints related to the agents are included in the form of indicator functions.

Note that consensus problem (<ref>) involves only one block of decision variables that is shared by the agents. This does not correspond to the practice of ADMM or its variants which essentially explore the problem decomposition across decision blocks for distributed optimization.  To address such an issue,  ADMM variants for consensus optimization (<ref>) are generally developed based on certain equivalent multi-block reformulations.
There are four widely-used reformulations indicated by (<ref>) - (<ref>).
A number of ADMM variants have been developed for both convex and nonconvex optimization by applying classical ADMM to those reformulations.
An overview of the relationships of the ADMM variants for solving consensus problem (<ref>) is shown in Fig. <ref>.
We distinguish the convex and nonconvex methods by solid and dashed boxes.  These methods are often preferred for different features, such as parallel implementation, networked communication schemes, low per-iteration complexity, random agent activation, and asynchronous computing (robust to communication delays or losses).

An overview of the relationships of the ADMM variants for solving consensus problem (<ref>) (the solid and dashed boxes to indicate convex and nonconvex methods respectively).

In the sequel, we introduce each of those methods. Particularly, we first present the four commonly-used reformulations which are helpful to understand the main ideas of related ADMM variants.   The reformulations all depend on
introducing local decision copies for individual agents and then enforcing  the consistency
of the decision copies. Their major differences lie in the ways of enforcing  consistency constraints.
Specifically for problem (<ref>), one natural and commonly-used  reformulation  is

\begin{align}
\label{pp:p4_v1} ~& \min_{\x:=(\x_i)_{i=0}^n}  F(\x) = \sum_{i = 1}^n f_i(\x_i)  \tag{$\PP4$-1}\\
&{\rm s.t.}~  \x_i = \x_0, ~ \forall i \in N. \notag
\end{align}

where $\x_i \in \R^{n}$ represents the local decision copy held by agent $i$. To guarantee the consistency of  decision copies, a global decision copy  $\x_0$  is introduced, which is  managed by a central coordinator. By enforcing  the consistency  of all local  copies to the global one, i.e.,  $\x_i = \x_0, \forall i \in N$, we directly have the  equivalence of   (<ref>) and  (<ref>).

In reformulation (<ref>),  the agents are assumed to communicate via a central coordinator. There exist cases that a system is fully decentralized and a central coordinator  does not exist. In such context,  the agents are restricted to communicate over a network or a graph. We refer to that as networked communication.  One direct reformulation that accounts for the networked communication is

\begin{align}
\label{pp:p4_v4} ~  &  \min_{\x:=(\x_i)_{i=1}^n} F(\x) =\sum_{i = 1}^n f_i(\x_i)   \tag{$\PP4$-2}\\
& {\rm s.t.~} \x_i = \x_j, \forall (i, j) \in E, i< j.  \notag
\end{align}

where $\x_i \in \R^n$ denotes the local decision copy held by agent $i$; $E$ indicates the edges of  communication network. To ensure the consistency of decision copies, a hard equality constraint $\x_i = \x_j$ is imposed for each pair of connected agents, i.e., $(i, j) \in E$. Since we only consider an undirected communication network,  one consistency constraint is sufficient for each pair of connected agents  and we therefore have $i < j$ in the reformulation.

When  the agents are restricted to communicate over a network or a graph,  another  reformulation is

\begin{align}
\label{pp:p4_v2} ~ & \min_{\x:=(\x_i)_{i=1}^n, \tbold:=(t_{ij})_{i, j  = 1}^n} ~ F(\x) = \sum_{i=1}^n f_i(\x_i)  \tag{$\PP4$-3}\\
& {\rm s.t.}~~ \x_i = t_{ij}, \quad \forall j \in N_i, i \in N. \notag\\
& \quad \quad \x_j = t_{ij}, \quad \forall j \in N_i, i \in N. \notag
\end{align}

where $\x_i \in \R^n$ denotes the local decision copy held by agent $i$ and  $N_i$ denotes the set of its neighbors (not including itself). To ensure the consistency of decision copies across the agents, a linking variable  $t_{ij} \in \R^n$  (also known as slack variable) is introduced for each pair of connected agents (i.e., $j \in N_i$ and $i \in N_j$). It is easy to show  that (<ref>) is equivalent to (<ref>) for any connected communication network.

Another similar reformulation  accounting  for networked communication scheme is

\begin{align}
\label{pp:p4_v3} ~  & \min_{\x:=(\x_i)_{i=1}^n, \z: =(\z_{ei}, \z_{ej})_{e \in E}} F(\x) = \sum_{i=1}^n f_i(\x_i)  \tag{$\PP4$-4}\\
&{\rm s.t.~} \x_i = \z_{ei}, ~\x_j = - \z_{ej},  \z_{ei} + \z_{ej}= \bm{0}, \forall e \in E.  \notag
\end{align}

where $\x_i \in \R^n$ denotes the local decision copy held by agent $i$; $E$ denotes the edges of the communication network, i.e., if  agent $i, j$ are neighbors, we have $(i, j) \in E$.  To ensure the consistency of decision copies held by the agents, two slack variable $\z_{ei} \in \R^n$ and $\z_{ej} \in \R^n$ are introduced for each edge $e \in E$ that connects agent $i, j$. Clearly, formulation (<ref>) is equivalent to (<ref>) for any connected communication network.

Note that reformulations (<ref>)-(<ref>) all correspond to  multi-block linearly constrained optimization where the objective is the sum of private objectives held by the agents. The ADMM variants to be discussed  are exactly developed based on the above reformulations. Most of them are the results of  the direct application of classical ADMM to the reformulations. However, different problem structures  were fruitfully exploited to favor easier implementations.

§.§.§ Classical ADMM
To account for networked communication scheme, [119] studied the direction application of classical ADMM  to  reformulation (<ref>). Note that problem  (<ref>) can be treated as a multi-block optimization with decision variables $\x_i, i \in N$.  The application of classical ADMM to solve multi-block (<ref>) is standard and  takes the following iterative scheme.

\begin{align*}
&\textbf{Classical ADMM:} ~~\\
&\textbf{Primal update:}~ \\
&~~\x_i^{k + 1} = \arg \min_{\x_i} \Lag_{\rho}\big(\x_i, (\x_j^{k+1})_{j <i}, (\x_j^k)_{j >i}, (\boldlambda_{ji}^k)_{j < i}, (\boldlambda_{ij})^k_{i<j}\big), \\
& \quad \quad \quad \quad \quad \quad \quad \quad \quad \forall i \in N. \\
&  \textbf{Dual update:} \\
&~~\lambda_{ij}^{k + 1} = \lambda_{ij}^k + \rho(\x_i^{k+1}  - \x_j^{k+1}), \forall (i, j) \in E, i < j. \\
\end{align*}

As discussed, the direct extension of classical ADMM to multi-block optimization generally requires some  extra conditions (e.g., strongly convex)  to ensure convergence.  In line with the previous results,  [119] established the convergence and   $\mathcal{O}(1/k)$  convergence rate of the method for a class of problems with  strongly convex objective functions $f_i$. Note that the method applies to other consensus problems (<ref>) that satisfy the necessary conditions of  multi-block extension of classical ADMM discussed in Section VI-B.

§.§.§ Consensus ADMM
Consensus ADMM results from the direct application of classical ADMM to reformulation (<ref>) [107].  Note that problem (<ref>) corresponds to a two-block optimization by treating the collections $\x:=(\x_i)_{i=1}^n$ and $\tbold: = (t_{ij})_{i, j =1}^n$ as two decision  blocks. When the problem is convex, classical ADMM is directly applicable. For the first block $\x:=(\x_i)_{i=1}^n$, we have the subproblem naturally decomposable across the decision components $\x_i, i \in N$, and for the second block $\tbold: = (t_{ij})_{i, j  = 1}^n$, we have  quadratic subproblems that admit  closed-form solutions.
By leveraging  the closed-form solutions of the slack variables $\tbold$, we have the succinct  implementation of consensus ADMM below.

\begin{align*}
&\textbf{Consensus ADMM:} ~~\\
&\textbf{Primal update:}~ \\
&  \x_i^{k+1} = \min_{\x_i} f_i(\x_i) + (\boldlambda_i^{k})^\top \x_i + \rho \sum_{j \in N_i} \Big\Vert \x_i \!-\!\frac{\x_i^k - \x_j^k}{2}\Big\Vert^2, \forall i \in N. \\
& \textbf{Dual update:}~   \boldlambda_i^{k+1} = \boldlambda_i^k + \rho \sum_{j \in N_i} (\x_i^{k+1} - \x_j^{k+1}), \forall i \in N.
\end{align*}

One salient feature of consensus ADMM is the parallelizable implementation. Specifically,  the agents are allowed to perform primal updates in a parallel  manner.  In addition, the method can accommodate a networked communication scheme. Since Consensus ADMM is a direct application of classical ADMM to a two-block convex optimization, its convergence and convergence rate directly follow  classical ADMM.

§.§.§ Inexact consensus ADMM
Consensus ADMM requires to solve the subproblems exactly at each iteration.  The could be expensive or not efficient when the objective functions $f_i$ are complex. To reduce per-iteration complexity,  inexact consensus ADMM was proposed as an inexact version of consensus ADMM  [107]. The key idea to linearize the subproblems with consensus ADMM and perform an inexact update at each iteration. The method takes the following iterative scheme.

\begin{align*}
&\textbf{Inexact consensus ADMM:} ~~\\
&\textbf{Primal update:}~   \\
&\quad \x_i^{k+1} = \min_{\x_i} \langle \nabla f_i(\x_i^k),  \x_i - \x_i^k \rangle + (\boldlambda^{k}_i)^\top \x_i\\
&\quad \quad \quad \quad + \rho \sum_{j \in N_i} \Big\Vert \x_i - \frac{\x_i^k - \x_j^k}{2}\Big\Vert^2 + \frac{\tau_i}{2}\Vert \x_i - \x_i^k \Vert^2, \forall i \in N. \\
& \textbf{Dual update:}~  \boldlambda_i^{k+1} = \boldlambda_i^k + \rho \sum_{j \in N_i} (\x_i^{k+1} - \x_j^{k+1}), \forall i \in N.
\end{align*}

Note that objective function $f_i$ in the primal update is replaced by  $\langle \nabla f_i(\x_i^k),  \x_i - \x_i^k \rangle + \frac{\tau_i}{2}\Vert \x_i - \x_i^k\Vert^2$ which is a local first-order approximation of $f_i$ plus a proximal term (the constant $f_i(\x_i^k)$ is discarded). It is easy to note that the resulting subproblems of the primal update admit closed-form solutions and yield low per-iteration complexity.
The convergence of inexact consensus ADMM was proved  for a class of problems:  $f_i(\x_i) = \phi_i(\A_i \x_i)$ where $\phi_i(\x)$  are strongly convex and $\Lips$ differentiable ($\A_i$ can be rank deficient)  [107]. Particularly, some convex (possibly nonsmooth) objective functions $g_i(\x_i)$ are admissible and  the subproblems should be adapted to $\x_i^{k+1} = \min_{\x_i} g_i(\x_i) + \langle \nabla f_i(\x_i^k),  \x_i - \x_i^k \rangle + (\boldlambda^{k}_i)^\top \x_i + \rho \sum_{j \in N_i} \big\Vert \x_i - \frac{\x_i^k - \x_j^k}{2}\big\Vert^2 + \frac{\tau_i}{2}\Vert \x_i - \x_i^k \Vert^2$,  which are proximal linearized subproblems.

§.§.§ Flexible ADMM
Flexible ADMM results from the direction application of classical ADMM to reformulation (<ref>) together with the deployment of a random agent activation mechanism  [115, 120]. Note that (<ref>) is a two-block optimization if  $\x_0$ and  $\x: =(\x_i)_{i=1}^n$ are treated as two decision blocks. This corresponds to the problem structures of classical ADMM.  The random agent activation mechanism means that  the computing agents are activated in a random manner in the iterative process.  This is often preferred when the computing agents have limited energy or suffer certain random ON/OFF behaviors.  Flexible ADMM takes the following iterative scheme.

\begin{align*}
&\textbf{Flexible ADMM:} ~~\\
& \textbf{Primal update:}~ \\
& \x_0^{k+1} \!\!=\! \begin{cases}
\arg\min\limits_{\x_0} \Lag_{\rho}(\x_0, (\x_i^{k})_{i=1}^n, (\boldlambda_i^k)_{i=1}^n),  \!\!\!\!& \text{if agent $0$ activated} \\
\x_0^k,  \!\!\!\!& \text{if agent $0$ not activated.} \\
\end{cases}\\
& \x_i^{k+1} \!=\! \begin{cases}
\arg\min\limits_{\x_i} \Lag_{\rho}(\x_0^{k+1}, \x_i,\boldlambda_i^k), &\quad \quad~~ \text{if agent $i$ activated} \\
\x_i^k,  & \quad \quad~~ \text{if agent $i$ not activated}
\end{cases}\\
&\textbf{Dual update:}~   \boldlambda_i^{k+1} = \boldlambda_i^k + \rho (\x_i^{k+1} - \x_0^{k+1}), i \in N.
\end{align*}

Note that the primal update of the second block $\x:=(\x_i)_{i=1}^n$ is performed in a distributed and parallel manner due to the naturally decomposable problem structure.
Similar to other flexible ADMM variants, the convergence of the method depends on the random agent activation mechanism. For the two usual cases that  each agent is activated either with a positive probability at each iteration or in an essentially cyclic manner (i.e., each agent is activated at least once within $K$ iterations),  [115, 120] established the convergence of the method for  nonconvex optimization  with $\Lips$ differentiable objective functions $f_i$.   To be noted,  the method directly applies to the convex counterpart since it is a direct application of classical ADMM [121]. Particularly, to improve communication efficiency,  [121] proposed a group-based communication mechanism for the method where the computing nodes are grouped and the information of nodes are  aggregated within groups before communication. Moreover,  for convex optimization,  the two decision blocks $\x_0$  and  $\x: = (\x_i)_{i=1}^n$  are interchangeable  as the first and second block. This is different from the  nonconvex optimization where only $\x:=(\x_i)_{i=1}^n$ can work as the last block. This is because we required a well-behaved last block  that satisfies certain $\Lips$ differentiable and rank properties to ensure convergence in nonconvex setting (see the discussions in Section IV-C).

§.§.§ Flexible linearized  ADMM
As an advanced version, flexible linearized ADMM was  proposed with the purpose of reducing per-iteration complexity with flexible ADMM  [115]. The basic idea is to linearize smooth subproblems and perform an inexact update instead of solving the subproblems exactly at each iteration. Flexible linearized ADMM takes the following iterative scheme.

\begin{align*}
&\textbf{Linearized flexible ADMM:} ~~\\
&\textbf{Primal update:}~   \\
&\x_0^{k+1} \!=\!
\begin{cases}
\arg\min\limits_{\x_0} \Lag_{\rho}(\x_0, (\x_i^k)_{k=1}^n, \boldlambda^k), ~~\text{if agent $0$  activated.}& \\
\x_0^k, \quad \quad \quad \quad \quad  \quad \quad \quad \quad \quad ~~~\text{if agent $0$  not activated.} &
\end{cases}\\
&\x_i^{k+1} = \begin{cases}
\arg \min\limits_{\x_i}\langle \nabla f_i(\x_0^{k+1}), \x_i^k, \boldlambda_i^k), \x_i - \x_i^k \rangle  + \langle \boldlambda_i^k, \x_i\rangle  & \\
\quad  + \frac{\tau_i}{2}\Vert \x_i - \x_i^k\Vert^2, \quad \quad \quad~~~~~ \text{if agent $i$ activated.} &  \\
\x_i^k, \quad \quad \quad \quad \quad  \quad \quad \quad \quad \quad   ~\text{if agent $i$ not activated.} &
\end{cases}\\
&\textbf{Dual update:}~   \boldlambda_i^{k+1} = \boldlambda_i^k + \rho (\x_i^{k+1} - \x_0^{k+1}), \forall i \in N.
\end{align*}

Note that the objective functions $f_i$ are linearized at $\x_i = \x_0^{k+1}$ and some proximal terms  $\frac{\tau_i}{2}\Vert \x_i - \x_i^k\Vert^2$ are added to the subproblems to control the  linear approximation accuracy. The convergence of flexible linearized ADMM was established under the same conditions as with flexible ADMM, i.e., the objective functions $f_i$ are $\Lips$ differentiable [115]. Note that we can also choose to linearize $f_i$ at $\x_i = \x_i^k$ and the theoretical proof  can be adapted accordingly.

§.§.§ Edge-based flexible ADMM
Edge-based flexible ADMM is another ADMM variant that adopts a random agent activation mechanism  [122]. The major difference  from flexible ADMM and its linearized version  is that the agents are randomly activated by the edges of their communication network. Specifically at each iteration, a subset of the edges of the communication network are randomly selected  and  the  agents connected to those edges will be activated  to perform  new updates. This is often referred to random edge-based agent activation mechanism. Edge-based flexible ADMM can be regarded as the result of applying classical ADMM to the reformulation (<ref>) together with the deployment of the random edge-based agent activation mechanism. Note that problem  (<ref>) corresponds to a two-block optimization by viewing the collections  $\x:=(\x_i)_{i=1}^n$ and $\z:=(\z_{ei}, \z_{ej})_{(i, j) \in e, e \in E}$ as two decision blocks.   Edge-based flexible ADMM takes the general iterative framework

\begin{align*}
& \textbf{Edge-based flexible ADMM:} ~~\\
& \text{if edge $e \in E$ is selected, the corresponding agents $(i, j)\in e$}\\ & \text{will perform the following primal and dual updates.} \\
& \textbf{Primal update:}~\\
& \begin{cases}
\x_i^{k+1} = \arg \min_{\x_i} \Lag_{\rho}(\x_i, \z_{ei}^k, u_e^k) \\
\x_j^{k+1} = \arg \min_{\x_j} \Lag_{\rho}(\x_j, \z_{ej}^k, v_e^k) \\
\end{cases} \\
& (\z_{ei}^{k+1}, \z_{ej}^{k+1})  = \begin{cases}
\arg \min_{\z_{ei}, \z_{ej}} \Lag_{\rho}(\x_i^{k + 1}, \x_j^{k + 1}, \z_{ei}, \z_{ej},  u_e^k, v_e^k)\\
{\rm s.t.~} \z_{ei} + \z_{ej} = 0. \\
\end{cases}\\
&\textbf{Dual update:}~  u_{e}^{k+1} = u_e^k + \rho(\x_i^{k+1} - \z_{ei}^{k+1}) \\
& \quad \quad \quad \quad \quad \quad v_{e}^{k+1} = v_e^k + \rho(\x_j^{k+1} + \z_{ej}^{k+1}) \\
\end{align*}

where $u_{e} \in \R^n$ and $v_{e} \in \R^n$ are Lagrangian multipliers associated with the consistency constraints $\x_i = \z_{ei}, ~\x_j = - \z_{ej}$ in the reformulation (<ref>).

One may note that the subproblems related to  decision variables $\z_{ei}$ and $\z_{ej}$ are quadratic and admit close-form solutions: $\z_{ej}^{k+1} = -\frac{u_e^k + v_e^k}{2\rho} - \frac{1}{2}(\x_i^{k+1} + \x_j^{k+1})$ and $\z_{ei}^{k+1} = \frac{u_e^k + v_e^k}{2\rho} + \frac{1}{2}(\x_i^{k+1} + \x_j^{k+1})$, which can be easily derived based on KKT conditions.
Similar to other flexible ADMM variants,  edge-based flexible ADMM   enables flexible implementation and is often preferred  when the agents have limited energy. The convergence of the method also depends on the  random edge-based agent activation mechanism. For the usual case where each edge is activated with a positive probability at each iteration, the convergence of the method towards global optima was established for convex optimization  [122]. Further, an $\mathcal{O}(1/k)$ convergence rate that characterizes the difference of objective gap and the feasibility violation under expectation was established.

§.§.§ Async-ADMM
Note that the above ADMM  variants are all implemented in a synchronous manner. The synchronization is in two senses: i) each agent  waits for the  information of its all interconnected agents to perform a new update [Flexible ADMM variants are slightly different due to the random agent activation mechanism.], and  ii)  each agent  uses the updated information of  its interconnected agents to perform a new update.   In practice, there often exist communication delays or losses caused by multiplied factors, such as limited network bandwidth, diverse processor configurations or inhomogeneous agent workloads, etc. In such context, the computing agents of a synchronous ADMM may have to wait a long time for the updated information of their interconnected agents over the network, thus disrupting or degrading the
the computation efficiency and scaling property of the method.
To address such an issue,  a number of works have studied  the asynchronous implementation of  ADMM variants. In contrast, an asynchronous implementation  means that i) each agent only waits for the information of its partial interconnected agents to perform a new update, and ii) each agent is allowed to use some outdated information of its interconnected agents to perform a new update.  As some typical examples, [123, 124, 125, 126] studied the asynchronous implementation of flexible ADMM. In flexible ADMM, the agents are distinguished  by one master and $n$ workers.  The master refers to the central coordinator  managing the global decision copy $\x_0$ and the workers are  agents updating the local decision copies $\x_i, \forall i \in N$.  Note the communications of the method are restricted to the  master  and  the  $n$ workers.  An synchronous implementation defines that the  master waits for  the  information of  all workers to proceed a new update. Clearly, when the number of works  is large, the master may have to wait a long time due to the possible communication delays or losses. As a remedy, [123, 124, 125, 126] proposed to allow the master to perform a new update only with  the information of  partial workers. Moreover, the information is not necessarily updated due to the lack of global synchronized iteration counter.  This leads to the asynchronous ADMM (async-ADMM) that takes the following iterative scheme.

\begin{align*}
&\textbf{Async-ADMM:} ~~\\
&\textbf{Primal update:}~ \\
& \x_i^k = \begin{cases}
\hat{\x_i}, & \text{if} ~i \in \mathcal{A}_k  \\
\x_i^k,  & \text{if} ~ i \in \mathcal{A}_k^c
\end{cases} \\
&  \boldlambda_i^k  = \begin{cases}
\hat{\boldlambda}_i, & \text{if} ~i \in \mathcal{A}_k  \\
\boldlambda_i^k,   & \text{if} ~ i \in \mathcal{A}_k^c\\
\end{cases}\\
& d_i = \begin{cases}
0, & \text{if} ~i \in \mathcal{A}_k  \\
d_i + 1,  & \text{if} ~ i \in \mathcal{A}_k^c
\end{cases} \\
&\!\!\! \text{Wait until  $\vert \mathcal{A}_k\vert \geq S$ and $d_i \leq \tau -1$:} \\
& \!\!\!\quad \x_0^{k+1} =\arg\min\limits_{\x_0} \Lag_{\rho}(\x_0, (\x_i^k)_{k=1}^N, \boldlambda^k),  \\
& \!\!\!\text{Send $\x_0^{k+1}$ to the nodes $\mathcal{A}_k$.} \\
\end{cases}\\
&\text{Worker}~i: \begin{cases}
&\!\!\!\text{Wait until $\hat{\x}_0 $ arrives}:\\
&\!\!\! \x_i^{k_i+1} = \arg\min\limits_{\x_i \in \X_i} \Lag_{\rho}(\hat{\x}_0, \x_i,  \boldlambda_i^{k_i})\\
\end{cases}, ~\forall i \in N. \\
& \textbf{Dual update:}~  \boldlambda^{k_i+1}_i = \boldlambda_i^{k_1} + \rho (\x_i^{k_i+1} - \hat{\x}_0), ~\forall i \in N.
\end{align*}

where the decision variables with a hat (i.e., $\hat{\x}_i$, $\hat{\boldlambda}_i$ and $\hat{\x}_0$) denote the new information received  from related agents by the master. Since the agents behave in an asynchronous manner,  each agent holds a private iteration counter.  We denote the iteration counter of master and worker by $k$ and $k_i, i \in N$, respectively.
In async-ADMM,  the master and workers behave in a slightly different manner due to the communication scheme.

For the master,   the workers are distinguished by whether their new updates have arrived or not at each iteration $k$. Specifically,  $\mathcal{A}_k$ and $\mathcal{A}_k^c$  indicate  the sets of workers whose  new updates have arrived or not, respectively. To ensure convergence, the master will set a maximum tolerable delay  bound  $\tau$ for each individual worker. This states that  at least one new update  must be served by each individual worker within  $\tau$ successive iterations of the master.  In other words, the information of the workers used by the master can be at most $\tau$ outdated. To enforce the maximum tolerable delay constraints, a delay counter $d_i$ is set for each individual worker.
At each iteration, if a new update  is  received from worker $i$, the delay counter will be cleared (i.e., $d_i = 0$), otherwise it will be increased by $1$ (i.e., $d_i = d_i + 1$).
In async-ADMM, instead of waiting for  the new updates of all workers to arrive, the master will proceed a new update with the new information of only partial workers. This is referred to  partial synchronization. A  specific partial synchronization mechanism $S$ states that the master will proceed  a new iterate after receiving  the new updates from any  $S$ workers and confirming that the remaining workers are not to exceed  the maximum tolerable delay bound $\tau$. For the special case that certain remaining workers are to exceed the bound $\tau$ ($d_i = \tau - 1$),   the master will hold on and wait for  the   new updates of those workers.  Note that when we have $S = n$ and $\tau = 1$, async-ADMM reduces to synchronous ADMM.   In the algorithmic implementation, we use $\x_i^k$ and $\boldlambda_i^k$ to denote the latest updates of worker $i$ hold by the master at iteration $k$. To be noted, they may not be the latest update of the workers due to the communication delays or losses. Note that if a new update $\hat{\x}_i$ is received from worker $i$, $\x_i^k$ will be updated accordingly, i.e.,  $\x_i^k = \hat{\x}_i$ by the master, otherwise the old updates will be hold to perform a new update.

For the worker, the update scheme is slightly different.  Since each worker only requires  information from the single master, each worker $i$ will wait  the new information of the mater to perform a new update.

The convergence of async-ADMM has been studied for both convex [123, 124] and nonconvex  [125, 126] optimization.
One  common and important condition to ensure convergence  is that the communication delay is bounded which states that a new  update from each individual worker must be served to  the master with a limited number of iterations (i.e., $\tau$). For convex optimization, [123] established the worst-case iteration complexity  $\mathcal{O}(\frac{n\tau}{k S})$. We imply that the convergence rate of async-ADMM closely relates to the problem scale $n$, the maximum tolerable  delay bound configuration $\tau$, and the partial synchronization mechanism $S$. There  usually exists  a trade-off between the iteration complexity (the number of iterations) and the waiting time of async-ADMM. Specifically, a larger tolerable delay bound $\tau$ and smaller synchronization $S$ often lead to less waiting time but more iterations to converge and vice verse.
For the special convex case where $f_i$ are strongly convex and $\Lips$ differentiable, a linear convergence of async-ADMM was established in [125].

§.§.§ Summary
This section reviewed ADMM variants for solving consensus problem  (<ref>). A number of ADMM variants have been developed either for convex or nonconvex settings.  We report those ADMM variants in terms of  main assumptions, decomposition schemes (i.e., type), convergence properties, main features and references in TABLE <ref>. We see that these ADMM variants can be distinguished by their features.
Consensus ADMM and inexact consensus ADMM enables full parallel  computation and can accounts for networked communication. Flexible ADMM, flexible linearized ADMM and edge-based flexible ADMM allow a random agent activation mechanism to save the energy consumption of agents. Async-ADMM enables asynchronous computing and is robust to communication delays or losses.
Inexact consensus ADMM and flexible linearized ADMM  are advantageous with low  per-iteration complexity.  Particularly,  we conclude that the ADMM variants for nonconvex consensus problem (<ref>)  require the objective functions $f_i$ to be $\Lips$ differentiable. This is in line with our discussions on the necessary conditions of nonconvex optimization in Section IV-C.

ADMM variants for solving consensus problem (<ref>)

Methods      Main assumptions      Types     Convergence     Features       References

[l]Classical ADMM       [l]$f_i$ strongly convex.       Gauss-Seidel      [l]Global convergence.
Global optima.
Convergence rate $\mathcal{O}(1/k)$.      [l]Networked communication.
Convex.     [119]

Consensus ADMM       [l]$f_i$ convex.        Gauss-Seidel     [l]Global convergence.
Global optima.
[l]Parallel implementation.
Networked communication.
Convex.     [107]

[l]Inexact consensus
ADMM       [l]$f_i(\x) = \phi_i(\A_i \x_i) + g_i(\x_i)$

$\phi_i$ strongly convex
and $\Lips$ differentiable.

$g_i$ convex (possibly nonsmooth).

$\A_i$ can be rank deficient.       Gauss-Seidel     [l]Global convergence.
Global optima.
[l]Parallel implementation.
Networked communication.
Low per-iteration complexity.
Convex.    [107]

Flexible ADMM       [l]$f_i$ $\Lips$ differentiable.        Gauss-Seidel     [l]Global convergence.
Stationary points.       [l]
Random agent activation.
Nonconvex     [115, 120]

linearized ADMM       [l]$f_i$ $\Lips$ differentiable.        Gauss-Seidel     [l]Global convergence.
Stationary points.      [l]Random agent activation.
Low per-iteration complexity.
Nonconvex.    [115]

flexible ADMM       [l]$f_i$ convex.         Gauss-Seidel     [l]Probability convergence.
Global optima.
Convergence rate $\mathcal{O}(1/k)$.       [l]Random agent activation.
Networked communication.
Convex.    [122]

Async-ADMM       [l]$f_i$ convex.
Bounded delay $\tau$.        Gauss-Seidel     [l]Global convergence.
Global optima.
Convergence rate $O(\tau/k)$.       [l]Robust to communication delays
(asynchronous computing).
Networked communication.
Convex.     [123, 124]

[l]$f_i$ $\Lips$ differentiable.
Bounded delay $\tau$.         Gauss-Seidel     [l]Global convergence.
Stationary points.       [l]Robust to communication delays.
(asynchronous computing)
Networked communication.
Nonconvex.    [125, 126]

§.§ Non-linearly constrained optimization

So far, we have focused on linearly constrained optimization. In other words, the couplings across the agents can be characterized by some  linear constraints. In practice, many other  systems  exist in which the interactions across the agents or subsystems are complex, leading to non-linear coupled constraints. Examples include but not limited to building thermal comfort management [23, 24] and power system control [127, 128].
This class of problems takes the general formulation of

\begin{align}
\label{pp:p5}  & \min_{\x= \{x_i\}_{i =1}^n}  \sum_{i=1}^n  f_{i}(x_{i}) \tag{$\PP5$}\\
\text{s.t.}~ & h_{i}(x_{i}, \{x_{j}\}_{j \in N_i}) = 0, \quad \forall i \in N. \notag\\
& g_{i}(x_{i}, \{x_{j}\}_{j \in N_i }) \leq  0, \quad \forall i \in N. \notag\\
& x_i \in X_i, \quad \quad \quad \quad \quad \quad ~\forall i \in N. \notag
\end{align}

where $f_i: \R^{n_i} \rightarrow \R$ are private objectives of the agents defined on their decision variables $x_i \in \R^{n_i}, i \in N$;  $h_i: \R^{\sum_{j \in N_i} n_j} \rightarrow \R$ and $g_i: \R^{\sum_{j \in N_i} n_j} \rightarrow \R$ characterize the non-linear (often nonconvex) couplings between agent $i$ and its neighbors $N_i$. The subset $X_i$ represents the local bounded convex constraints  of agent $i$. Problem (<ref>) models a class of multi-agent optimization where the agents hold private objectives but their  decisions  are coupled in  non-linear (often nonconvex) equality and inequality constraints. Currently, distributed methods to handle problem (<ref>) (often nonconvex) efficiently are extremely lacking.

Though ADMM variants for diverse classes of linearly constrained optimization are  available, extending them to non-linearly constrained problem (<ref>) (often with nonconvex couplings) is nontrivial due to the difficulty to establish convergence.  Currently, the  existing ADMM variants for solving problem (<ref>) are quite limited. To our best knowledge, the two-level ADMM proposed in [129]  has been the main solution with convergence guarantee. We discuss that method below.

§.§.§ Two-level ADMM
As a celebrated work, [129] studied a two-level ADMM for solving non-linearly constrained  problem (<ref>) in general nonconvex setting (i.e., $f_i$, $h_i$, $g_i$ are nonconvex). The basic  idea is to first convert problem (<ref>) into a  linearly constrained optimization and then explore the application of existing ADMM variants.  Though the idea is natural, there are some intrinsic challenges to be overcome, i.e., the two necessary conditions (see the discussions in Section IV-C) to ensure the convergence of ADMM and its variants in nonconvex setting can not be provided simultaneously. To overcome such a challenge, two-level ADMM
proposed the idea of introducing a block of slack variables as the last block.

The idea of this method should be  clear from below.

Specifically, by introducing decision copies for the interconnected agents, we have the following equivalent linearly constrained reformulation for problem (<ref>).

\begin{align}
\label{pp:p5-v1}  \min_{ \x = (\x_i)_{i=1}^n,  \hx}& F(\x) = \sum_{i=1}^n f_i(\x_i) \tag{$\PP5$-1} \\
\text{s.t.}~ & \sum_{i=1}^n \A_i \x_i + \B \hx = \bm{0}. \notag \\
&  h_i(\x_i) = \bm{0}, g_i(\x_i) \leq \bm{0},  \notag\\
&\x_i \in \X_i, \quad  i \in N. \notag \\
&\hx \in \bar{\mathbf{X}}. \notag
\end{align}

where $\mathbf{x}_i: = (x_{i}, (x_{j})_{j \in N_i})$ denotes the augmented decision variable held by agent $i$,  which consists of its own decision variable $x_i$ and the estimates $(x_{j})_{j \in N_i}$ for its neighbors indexed by subset $N_i$;  $\bar{\x}: =(x_i)_{i=1}^n$ represents a global copy of all decision variables over the network.
The coupled linear constraints $\sum_{i=1}^n \A_i \x_i + \B \bar{\x} = \bm{0}$ enforce the consistency of the decision copies over the network.  By defining $\A: = (\A_i)_{i=1}^n$ and  $\x: = (\x_i)_{i=1}^n$, the  coupled linear constraints can be  expressed by the compact form of  $\A \x + \B \bar{\x} = \bm{0}$; $\X_i: = \cup_{j \in N_i} X_j$ denotes the augmented local constraints for agent $i$.

Note that problem (<ref>) corresponds to a two-block optimization by treating the decision variables  $\x:=(\x_i)_{i=1}^n$ and $\bar{\x}$ as two decision blocks.   This corresponds to the problem structure of most existing ADMM variants. However, they are not applicable  due to the existence of nonconvexity.  As discussed in Section VI-C, we generally require a well-behaved last block $\y$ satisfying the two necessary conditions to ensure convergence:  i) the last decision block $\y$ is unconstrained and has $\Lips$ differentiable objective, and ii) ${\rm Im}(\A)\subseteq {\rm Im}(\B)$, $\B$ has full column rank or the mapping $H(\uu) = \{\arg \min_{\y} \phi(\x, \y): \B \y = \uu\}$ is unique and $\Lips$ smooth. Here  $\phi(\x, \y)$ refers to the objective of problem (<ref>),  $\B$ denotes the coefficient matrix of a last block $\y$ and $\A$ represents an augmented coefficient matrix for the decision blocks excluding $\y$.  Despite the direct linearly constrained reformulation (<ref>), it was argued that the two necessary conditions i)-ii) can not be provided simultaneously. Specifically, to fulfill condition ii), we should have $\x:=(\x_i)_{i=1}^n$ work as the last block (we actually have ${\rm Im } (\A) \supseteq {\rm Im} (\B)$ in (<ref>)).  However, the presence of local constraints (i.e., $\X_i, h_i, g_i$) makes the subproblems $\x$ nonsmooth, violating condition i). Oppositely, if we assume $\hx$  as the last block, we do not have condition ii).   To overcome such a challenge, a block of slack variables  $\z$  is introduced as a well-behaved last block. This leads to the following equivalent three-block linearly constrained reformulation.

\begin{align}
\label{pp:p5-v2}	 \min_{ \x: = (\x_i)_{i=1}^n,  \hx, \z} & F(\x) =\sum_{i=1}^n f_i(\x_i) \tag{$\PP5$-2} \\
\text{s.t.}~& \A \x + \B \hx + \z = \bm{0}. \notag \\
&  h_i(\x_i) = \bm{0}, g_i(\x_i) \leq \bm{0}, \notag\\
& \x_i \in \X_i, \quad  i  \in  N. \notag \\
&\hx \in \bar{\mathbf{X}}. \notag \\
&  \z = \bm{0}. \notag
\end{align}

On top that,  a  two-level ADMM variant  was proposed for solving (<ref>) in which  an ALM is used in the upper level to gradually force the slack variables to zero and the classical ADMM is used in the lower level to solve some relaxed multi-block optimization with the slack variables as the last block.
The method takes the following two-loop iterative scheme.

\begin{align*}
&\textbf{Two-level ADMM:} ~~\\
&\!\!\! \!\! \text{Outer-loop}  \begin{cases}
\!\!\! \!\!\!\text{Inner-loop} \begin{cases}
& \! \! \! \textbf{Primal update:}~ \\
&  \! \! \! \x^{k+1} \!=\! \arg \min_{\x} \Lag_{\rho}(\x, \bar{\x}^k, \z^k, \boldlambda^k, \bm{\gamma}^p)\\
& \! \!  \! \bar{\x}^{k + 1} \! =\! \arg \min_{\bar{\x}} \Lag_{\rho}(\x^{k+1}, \bar{\x}, \z^k, \boldlambda^k, \bm{\gamma}^p) \\
& \! \! \! \z^{k+1} \!=\! \arg \min_{\z}  \Lag_{\rho}(\x^{k + 1}, \bar{\x}^{k + 1}, \z, \boldlambda^k, \bm{\gamma}^p)\\
& \! \! \! \!\! \!\textbf{Dual update:} \\
&  \boldlambda^{k+1} \!=\! \boldlambda^k + \rho(\A \x^{k+1} + \B \bar{\x}^{k+1} + \z^{k+1}) \\
\end{cases} \\
& \!\!\!\textbf{Dual update:}~~ \bm{\gamma}^{p+1} = \bm{\gamma}^p + \beta \z^{k+1}
\end{cases}
\end{align*}

where $\boldlambda$  and $\bm{\gamma}$  are Lagrangian multipliers associated with the coupled linear constraints $\A \x + \B\bar{\x} +\z = \bm{0}$ and the hard constraints $\z =\bm{0}$; $\rho$ and $\beta$ are penalty parameters  for  the ADMM in  the lower level (Inner-loop)  and the ALM in the upper level (Outer-loop); $k$ and $p$ are related  iteration counters.
In the primal update of Inner-loop, we have the subproblems $\x:=(x_i)_{i=1}^n$  naturally decomposable across the decision components $\x_i, i \in N$ and thus can be performed in a distributed and parallel manner.
The convergence of the two-level ADMM towards stationary points was established for general nonconvex optimization [129].  Further, the efficacy of the method was demonstrated by a concrete  application to   power system control  [128].

Summary: This section reviewed ADMM variants for solving  non-linearly constrained problem  (<ref>) (often nonconvex). Currently, this related methods are quite limited. One solution with convergence guarantee  is the  two-level ADMM.   This is mainly caused by the intrinsic challenges that the two necessary conditions to ensure convergence of ADMM variants in nonconvex setting are not satisfied simultaneously. Two-level ADMM proposed the idea of introducing  slack variables to fulfill such  two conditions.  Though the efficacy of the method has been demonstrated both theoretically and empirically, the implementation is generally at the cost of high iteration complexity caused by the two-level iterative scheme.

§ DISCUSSIONS AND FUTURE RESEARCH DIRECTIONS

Based on the previous survey, this section discusses several important future research directions related to ADMM and its variants for distributed optimization.

§.§ Nonconvex extensions

From this survey,  we see  that ADMM and its variants for convex optimization have  been  studied extensively and broad results are now available for broad classes of problems.
However, for the more broad nonconvex counterparts,  the related solutions are still quite limited. Moreover, the existing ADMM variants for nonconvex optimization are all restricted to the two necessary conditions  [68, 129]: i)  the last block $\y$ is unconstrained and has $\Lips$ differentiable objective, and ii) ${\rm Im}(\A)\subseteq {\rm Im}(\B)$, $\B$ has full column rank or the mapping $H(\uu) = \{\arg \min_{\y} \phi(\x, \y), {\rm s.t.~} \B \y = \uu\}$ is unique and $\Lips$ smooth (see the discussions in Section IV-C).  There exist a wide spectrum of problems  that fail to satisfy such two conditions, such as problems (<ref>) and (<ref>). In such situations, it represents a big challenge to develop  an ADMM variant  with  convergence guarantee.

To figure out the solutions, we require to comprehensively understand the general framework of establishing convergence  for an ADMM or its variant in nonconvex setting. Specifically, the key step  is to identify a sufficiently decreasing and lower bounded Lyapunov function, which can indicate the convergent property of  generated sequences. The sufficiently decreasing and lower boundness properties of a Lyapunov  function for an ADMM or its variant  state that [68, 115]

\begin{align*}
&T(\x^{k+1}, \boldlambda^{k+1}) - T(\x^k, \boldlambda^k) \\
&\quad \quad \quad \quad \leq -\alpha_{\x}\Vert \x^{k+1} - \x^k\Vert^2 - \alpha_{\boldlambda} \Vert \boldlambda^{k+1} - \boldlambda^k\Vert^2, \\
& T(\x^k, \boldlambda^k) > -\infty,
\end{align*}

where  $T(\cdot, \cdot)$ denotes a general Lyapunov function;  $\{\x^k\}_{k=1}^K$  and $\{\boldlambda^k\}_{k=1}^K$ are primal and dual sequences;  $k$ is the iteration counter;  $\alpha_{\x}$ and $\alpha_{\boldlambda}$ are positive scalars. The  AL functions and their variants are often used as the Lyapunov functions (see [68] for example).
To identify such a Lyapunov function, we generally require to bound the dual updates  $\Vert \boldlambda^{k+1} - \boldlambda^k\Vert^2$ by the primal updates $\Vert \x^{k+1} - \x^k\Vert^2$. Conditions i)-ii) are exactly used to achieve such an objective.

In recent years, some interesting and promising efforts have been made to relax such two conditions.
Particularly, some works have been devoted to overcoming the challenges caused by  the lack of  $\Lips$ differentiable property of  last block.
For example, [110] proposed to employ a discounted dual update scheme to bound the dual updates manually  when the $\Lips$ differentiable property of last block is lacking. From another perspective,  [130] proposed the idea of using a smooth and $\Lips$ differentiable Moreau envelope to approximate a nonsmooth but weakly convex objective of last block. In terms of the conditions on the coefficient matrices of last block, [129, 131] proposed the idea of introducing slack variables as a well-behaved last block.  These works show that the restrictive conditions i)-ii) are possible to be relaxed by reinforcing the existing methods. This is important to further enhance the capability of ADMM and its variants for more broad distributed optimization and presents one of the important future research directions to be addressed.

§.§ ADMM acceleration

Though the numerical convergence depends on specific problems, classical ADMM and its many variants only promise an  $\mathcal{O}(1/k)$ convergence rate. For an iterative algorithm, the convergence rate usually characterizes the iteration complexity and communication burden of a method.  We often prefer a faster convergence rate  to approach an appropriate solution with less iterations and less communications.
To achieve such a goal,  some efforts have been made to accelerate the convergence of classical ADMM. A typical and successful  example is the fast ADMM [74, 73] which  is able to improve the convergence rate of classical ADMM by an order via the integration of Nesterov acceleration technique. However, this method only applies to a special class of problems that are strongly convex and with additional  quadratic structure.   The extension of fast ADMM  to  more general problems still remains an open issue to be addressed. This is nontrivial and faced with some  intrinsic challenges to be overcome. Specifically,  Nesterov acceleration is primarily developed for  first-order descent solvers, however  ALM and its inexact versions (e.g., the ADMM variants based on $\gauss$ and $\Jacobian$ decomposition) are basically not descent solvers. A descent solver states that we have the objective value  decrease along the iterations.  In terms of such issue,  some works argued that the descent property of an ALM or ADMM variant  can  be ensured by adding some monitoring and correction steps [74]. This actually sheds some light on generating fast ADMM to more general problems.

In addition to Nesterov acceleration, Anderson acceleration  also began to draw interest for accelerating ADMM or its variants [132, 133]. However, only some numerical results are currently available and the solid theoretical results regarding the convergence rate characterizations  remain to be established.

Overall, the above acceleration techniques follow a  general framework to accelerate ADMM or its variants. The key step  is to twist or modify the generated  primal and dual sequences in the iterative process, which can be uniformly characterized by $\hat{\w}^{k+1} = {\texttt{acc}}\big(\w^{k+1}, \w^k, \w^{k-1}, \cdots\big), k = 1, 2, \cdots$,  where $k$ is the iteration counter,  $\w: = (\x^\top, \boldlambda^\top)^\top$ is the stack of primal and dual variables,  $\w^k$ denotes the generated update by  an ADMM or its variant at iteration $k$,  and $\hat{\w}^k$ denotes the modified update for acceleration at iteration $k$. We use  to denote an acceleration technique, which often corresponds to a linear  combination. This general acceleration framework for ADMM and its variants has been documented in [134], however the theoretical convergence rate characterizations for such general acceleration framework  remain to be addressed. This represents another important and significant future research direction.

§.§ Asynchronous ADMM

From this survey, we note that most of the existing ADMM variants are synchronous. Specifically, they generally define or assume that the computing agents behave in a synchronous manner. This is in the sense that the agents share the same iteration counter and use the updated information of their interconnected agents to proceed new updates at each step.  This may be problematic considering the practice that one agent usually has no idea about the progress of  another in the iterative process.  Besides, the communications are often not idealized and suffer from communication delays, communication losses and temporary  disconnections.
When a synchronous ADMM is applied to such context, the computation efficiency or convergence property of the method could be largely degraded or disrupted due to the long waiting time or the failure of information synchronization over the networks.

It was demonstrated  by simulations that the asynchronous behaviors of computing agents caused by communication delays and  solver diversity could  lead to the oscillation of a synchronous ADMM [135]. To address such an issue, the concept of asynchronous ADMM is thus proposed with the idea  to account for the asynchronous behaviors of computing agents while securing the convergence property and computation efficiency of an ADMM or its variant.

As discussed before, some efforts have been made to develop asynchronous ADMM variants both for convex [123, 124] and nonconvex consensus  optimization [125, 126]. The main idea is that at each step, the coordinator agent only waits for the information of partial computing agents involved in the system to perform a new update with the objective to cut off  waiting time.

This is often referred to partial synchronization.  To ensure  convergence of the method, a critical step is to set a maximal tolerable delay  bound $\tau$ to each computing agent. This enforces that
the information of all agents used by the coordinator can be at most $\tau$ old. This method has already found many successful applications, such as the peer-to-peer energy trading matching in a deregulated electricity market  [136].

Though some notable results have been achieved along this direction, the study of  asynchronous implementation of ADMM variants is still at the very early stage. Specifically, most of the exiting asynchronous ADMM variants are specific to consensus optimization that adopts a master-worker communication scheme. For the other classes of problems discussed in this paper that also suffer from the asynchronous behaviors of computing agents,  asynchronous solutions are still extremely lacking. In addition, when  a network or a  graph defining the communication scheme across the agents exists, the interactions of computing agents with asynchronous behaviors could be much more complex. In such context, the existing asynchronous ADMM variants are not applicable and we are required to develop other effective solutions. In addition, there often exists a trade-off between the waiting time and the iteration complexity that are determined by the maximal bounded delay configurations and  partial synchronization mechanism. A smaller tolerable delay bound and larger partial synchronization mechanism  often yield low iteration complexity but lead to longer waiting time.   Oppositely, a larger tolerable  delay bound and smaller partial synchronization  often result in higher iteration complexity but shorter waiting time.  In practice, we are often concerned with the time efficiency of a solution method, therefore some theoretical results are required to guide how to make  a trade-off between the tolerable delay bound and partial synchronization to achieve the best time efficiency. These are the interesting topics to be studied in the future.

§.§ ADMM + Reinforcement Learning

Learning is one of the most important tools for developing machine intelligence in the big data era. As one of the most important branches of machine learning, reinforcement learning (RL) has been widely studied  and achieved widespread success for sequential decision making under dynamic and uncertain environments for both engineering and computer systems [137, 138].  RL and ADMM have emerged as the two most important tools for optimization and control, showing complementary benefits. Specifically,  RL is not restricted to system or problem complexity and only entails that  the system performance or problem objective can be evaluated. In contrast, ADMM and its variants provide a powerful and flexible general framework to  achieve reliable coordination of multi-agent computation, thus overcoming  the computation intensity faced by a centralized computation architecture. Considering the dual  benefits, the combination of RL and ADMM  is expected to enable more powerful control and optimization tools and shows wide  prospects.

In  very recent years, the combination of RL and ADMM has began to draw interest. The related works can be divided into two categories by their perspectives. One is to deploy ADMM into an existing RL framework to enable multi-agent reinforcement learning (MARL) or distributed reinforcement learning (DRL), and the other one is to integrate an RL agent into an existing ADMM  variant for handling complex subproblems.  As the first category, [139] studied the combination of ADMM and inexact ADMM with RL to enable DRL for a class of multi-agent systems where the agents have private local objectives but are coupled through a joint decision making scheme.

The proposed method enables the multiple agents to collaboratively learn a joint control policy in a distributed manner considering their individual objectives while not disclosing their rewards or preferences to the others.  For the same problem set-up, [140] proposed to combine a  stochastic ADMM with RL  by using the gradient information of system performance for policy updates. As the second category,  [141, 30]  studied the integration of RL agents into classical ADMM framework with the objective to overcome the computation challenges of solving complex and non-linear subproblems.

Currently, the combination of ADMM or its variants with RL is at the very beginning. Many critical issues require to be addressed. Typically, how to best integrate the two method to improve computation efficiency  is still under discussion.  In addition, it still remains a big challenge to establish the  theoretical convergence for their combinations due to the feature of RL which tries to improve policies under uncertainties instead of solving optimization problems comprehensively. This represents another important future research directions to be studied.

§.§ ADMM + Federated Learning

Federated learning (FL) is an emerging distributed  machine learning setting with the main idea of  empowering massive dispersed  clients to learn models by leveraging locally available data and  relying on a central server to aggregate the models to achieve coordination [142, 143]. Generally, FL provides the scope of training machine learning models over an edge-based distributed computation architecture.
This is in contrast to the traditional centralized learning paradigm where a central server is authorized to learn a comprehensive model independently and thoroughly  by  collecting data from the whole system.
With the proliferation and penetration of Internet of Things (IoT), data is generated at an unprecedented rate and in a geographically distributed manner.
Meanwhile, the whole society is raising awareness of data and information privacy.
This poses the issue of developing communication efficient and privacy-preserving  machine learning tools. FL is thus proposed with the objective to enable  high data privacy, low communication burden, and high computation efficiency. One distinguishing feature of FL is that the clients only communicate  with the server for model parameters and can keep their data private.

FL essentially corresponds to decomposing a mathematical optimization across a number of computing agents. More specifically, by viewing the model parameters as the decision variables, FL generally corresponds to solving a consensus optimization (possibly nonconvex) in a distributed manner.
As one of the most powerful  distributed methods,  ADMM has emerged  as a  popular tool for enabling FL. A number of works have studied ADMM variants for FL. Typical examples include [144, 145, 146].  Specifically, to reduce communication and computation burden, [144, 145] studied an ADMM variant with a flexible  communication mechanism for FL, i.e.,  the clients only communicate with the central server at fixed time points $k_0, 2k_0, \cdots$ instead of at each iteration, where $k_0$ denotes the interval of communication. To account for communication delays or losses as well as random client ON/OFF behaviors, [146] studied  an  ADMM variant with a  flexible client participation mechanism for FL, i.e., at each iteration only a subset  of the clients  are selected to update models [146].  In fact, these ADMM variants are exactly the specific implementation of the ADMM variants for consensus optimization  discussed before.

ADMM presents  an important distributed framework for FL considering its flexibility, robustness and efficiency. Specifically,  ADMM and its variants allow to  activate the computing agents in a flexible manner, such as the random agent activation and the partial agent participation.  Besides, ADMM and its variants rely on quite weak assumptions to ensure convergence and are applicable to broad classes of problems both in convex and nonconvex settings.  In addition, many ADMM variants are compatible with inaccurate or uncertain information as well  communication delays or losses.  Moreover, ADMM  variants generally  show  faster convergence over many other distributed methods.  Last but very important, most of the ADMM variants  allow to solve the subproblems inexactly with quite low per-iteration complexity. These are the favorable features of ADMM and its variants that are expected to be celebrated by FL.

However, FL is somehow different from mathematical optimization in the sense that  a learning or training agent is generally deployed to train a neural network instead of solving a mathematical subproblem with explicitly available model at each iteration. This poses the challenges to establish the convergence of the methods in the machine learning context.
In addition, FL often places higher emphasis on the computation and communication efficiency than the solution accuracy. This implies that the existing ADMM variants may have to be adapted or reinforced for FL. These represent some other important and emerging future research directions.

§ CONCLUSION

This paper provided a comprehensive survey on  ADMM and its variants for distributed optimization. We discerned the five major classes of problems that have been mostly concerned in the literature.  For each class of problems, we discussed the related ADMM variants  from the perspectives of main ideas,  main assumptions, decomposition scheme, convergence properties and main features.  Based on the survey, we further identified several important future research directions to be addressed, which include: i) the  extension of ADMM variants to  more general nonconex optimization, ii) the acceleration of ADMM variants with convergence and convergence rate characterizations. iii) the  asynchronous implementation of ADMM variants accounting for the asynchronous behaviors of computing agents, iv) the combination of ADMM and reinforcement learning (RL) for enabling more powerful optimization and control tools, v) the combination of ADMM and federated learning (FL) for efficient and privacy-preserving machine learning.   This review covered most of the developments of ADMM and its variants in  very recent decades and can work as a tutorial both for developing distributed optimization in broad areas and identifying  existing  research gaps.

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