A Proximal Dual Consensus ADMM Method for Multi-Agent Constrained Optimization

I Introduction

Multi-agent distributed optimization has been of great interest due to applications in sensor networks , cloud computing networks and due to recent needs for distributed large-scale signal processing and machine learning tasks . Distributed optimization methods are appealing because the agents access and process local data and communicate with connecting neighbors only , thereby particularly suitable for applications where the local data size is large and the network structure is complex. Many of the problems can be formulated as the following optimization problem

Various distributed optimization methods have been proposed in the literature for solving problems with the form of (P). For example, the consensus subgradient methods can be employed to handle (P) by solving its Lagrange dual problem . The consensus subgradient methods are simple to implement, but the convergence rate is slow. In view of this, the alternating direction method of multipliers (ADMM) has been used for fast distributed consensus optimization . Specifically, the work proposed a consensus ADMM (C-ADMM) method for solving a distributed LASSO problem. The linear convergence rate of C-ADMM is further analyzed in , and later, in , C-ADMM is extended to that with asynchronous updates. By assuming that a certain coloring scheme is available to the network graph, the works in proposed several distributed ADMM (D-ADMM) methods for solving problems with the same form as (P). The D-ADMM methods require each agent either to update the variables sequentially (not in parallel) or to solve a min-max (saddle point) subproblem at each iteration. In the recent work , the authors proposed a distributed optimization method, called dual consensus ADMM (DC-ADMM), which solves (P) in a fully parallel manner over arbitrary networks as long as the graph is connected. An inexact counterpart of DC-ADMM was also proposed in for achieving a low per-iteration complexity when $F$ is complex.

In this paper, we improve upon the works in by presenting new computationally efficient distributed optimization methods for solving (P). Specifically, due to the presence of the polyhedra constraints ${\bm{C}}_{i}{\bm{x}}_{i}\preceq{\bm{d}}_{i}$ in (1e), the agents in the existing methods have to solve a polyhedra constrained subproblem at each iteration. Since projection onto the polyhedra constraint is not trivial, closed-form solutions are not available and, moreover, simple algorithms such as the gradient projection method cannot handle this constrained subproblem efficiently. To overcome this issue, we propose in this paper a proximal DC-ADMM (PDC-ADMM) method where each of the agents deals with a subproblem with simple constraints only, which is therefore more efficiently implementable than DC-ADMM. This is made possible by the use of the proximal minimization method [14, Sec. 3.4.3] to deal with the dual variables associated with the polyhedra constrains, so that the constraints can be softly handled as penalty terms in the subproblems. Our contributions are summarized as follows.

We propose a new PDC-ADMM method, and show that the proposed method converges to an optimal solution of (P) with a worst-case $\mathcal{O}(1/k)$ convergence rate, where $k$ is the iteration number. Numerical results will show that the proposed PDC-ADMM method exhibits a significantly lower computation time than DC-ADMM in .

We further our study by presenting a randomized PDC-ADMM method that is tolerable to randomly ON/OFF agents and robust against imperfect communication links. We show that the proposed randomized PDC-ADMM method is convergent to an optimal solution of (P) in the mean, with a worst-case $\mathcal{O}(1/k)$ convergence rate.

The rest of this paper is organized as follows. Section II presents the applications, network model and assumptions of (P). The PDC-ADMM method and the randomized PDC-ADMM method are presented in Section III and Section IV, respectively. Numerical results are presented in Section V and conclusions are given in Section VI.

Notations: ${\bm{A}}\succeq{\bm{0}}$ ($\succ{\bm{0}}$) means that matrix ${\bm{A}}$ is positive semidefinite (positive definite); ${\bm{a}}\preceq{\bm{d}}$ indicates that $({\bm{d}})_{i}-({\bm{a}})_{i}\geq 0$ for all $i$, where $({\bm{a}})_{i}$ means the $i$th element of vector ${\bm{a}}$. ${\bm{I}}_{K}$ is the $K\times K$ identity matrix; ${\bf 1}_{K}$ is the $K$-dimensional all-one vector. $\|{\bm{a}}\|_{2}$ denotes the Euclidean norm of vector ${\bm{a}}$, $\|{\bm{a}}\|_{1}$ represents the 1-norm, and $\|{\bm{x}}\|^{2}_{\bm{A}}\triangleq{\bm{x}}^{T}{\bm{A}}{\bm{x}}$ for some ${\bm{A}}\succeq{\bm{0}}$; ${\rm diag}\{a_{1},\ldots,a_{N}\}$ is a diagonal matrix with the $i$th diagonal element being $a_{i}$. Notation $\otimes$ denotes the Kronecker product. $\lambda_{\max}({\bm{A}})$ denotes the maximum eigenvalue of the symmetric matrix ${\bm{A}}$.

II Applications, Network Model and Assumptions

Problem (P) has applications in machine learning , data communications and the emerging smart grid systems , to name a few. For example, when $f_{i}({\bm{x}}_{i})=\|{\bm{x}}_{i}\|_{2}^{2}$ $\forall i$, (P) is the least-norm solution problem of the linear system $\sum_{i=1}^{N}{\bf E}_{i}{\bm{x}}_{i}={\bm{q}}$; when $f_{i}({\bm{x}}_{i})=\|{\bm{x}}_{i}\|_{1}$ $\forall i$, (P) is the well-known basis pursuit (BP) problem ; and if $f_{i}({\bm{x}}_{i})=\|{\bm{x}}_{i}\|_{2}$ $\forall i$, then (P) is the BP problem with group sparsity . The LASSO problem can also be recast as the form of (P). Specifically, consider a LASSO problem with column partitioned data model [17, Fig. 1],,

where ${\bm{A}}_{i}$’s contain the training data vectors, ${\bm{b}}$ is a response signal and $\lambda>0$ is a penalty parameter. By defining ${\bm{x}}_{0}\triangleq\sum_{i=1}^{N}{\bm{A}}_{i}{\bm{x}}_{i}-{\bm{b}}$, one can equivalently write (2) as

where ${\bm{x}}_{0}$ is a slack variable and $U$ is the cost function for power imbalance. Problem (4) is again an instance of (P).

II-B Network Model and Assumptions

We model the multi-agent network as a undirected graph $\mathcal{G}=\{{\mathcal{V}},\mathcal{E}\}$, where ${\mathcal{V}}=\{1,\ldots,N\}$ is the set of nodes (i.e, agents) and $\mathcal{E}$ is the set of edges. In particular, an edge $(i,j)\in\mathcal{E}$ if and only if agent $i$ and agent $j$ are neighbors; that is, they can communicate and exchange messages with each other. Thus, for each agent $i$, one can define the index subset of its neighbors as $\mathcal{N}_{i}=\{j\in{\mathcal{V}}\mid(i,j)\in\mathcal{E}\}$. Besides, the adjacency matrix of the graph $\mathcal{G}$ is defined by the matrix ${\bm{W}}\in\{0,1\}^{N\times N}$, where $[{\bm{W}}]_{i,j}=1$ if $(i,j)\in\mathcal{E}$ and $[{\bm{W}}]_{i,j}=0$ otherwise. The degree matrix of $\mathcal{G}$ is denoted by ${\bm{D}}={\rm diag}\{|{\mathcal{N}}_{1}|,\ldots,|{\mathcal{N}}_{N}|\}$. We assume that

The undirected graph $\mathcal{G}$ is connected.

Assumption 1 is essential for consensus optimization since it implies that any two agents in the network can always influence each other in the long run. We also have the following assumption on the convexity of (P).

(P) is a convex problem, i.e., $f_{i}$’s are proper closed convex functions (possibly non-smooth), and $\mathcal{S}_{i}$’s are closed convex sets; there is no duality gap between (P) and its Lagrange dual; moreover, the minimum of (P) is attained and so is its optimal dual value.

III Proposed Proximal Dual Consensus ADMM Method

In the section, we propose a distributed optimization method for solving (P), referred to as the proximal dual consensus ADMM (PDC-ADMM) method. We will compare the proposed PDC-ADMM method with the existing DC-ADMM method in , and discuss the potential computational merit of the proposed PDC-ADMM.

The proposed PDC-ADMM method considers the Lagrange dual of (P). Let us write (P) as follows

for all $i\in{\mathcal{V}}$. To enable multi-agent distributed optimization, we allow each agent $i$ to have a local copy of the variable ${\bm{y}}$, denoted by ${\bm{y}}_{i}$, while enforcing the distributed ${\bm{y}}_{i}$’s to be the same across the network through proper consensus constraints. This is equivalent to reformulating (6) as the following problem

where $\{{\bm{t}}_{ij}\}$ and $\{{\bm{s}}_{i}\}$ are slack variables. Constraints (8b) and (8c) are equivalent to the neighbor-wise consensus constraints, i.e., ${\bm{y}}_{i}={\bm{y}}_{j}~{}\forall j\in{\mathcal{N}}_{i},~{}i\in{\mathcal{V}}$. Under Assumption 1, neighbor-wise consensus is equivalent to global consensus; thus (8) is equivalent to (6). It is worthwhile to note that, while constraint (8d) looks redundant at this stage, it is a key step that constitutes the proposed method as will be clear shortly.

Let us employ the ADMM method to solve (8). ADMM concerns an augmented Lagrangian function of (8)

Equations (10), (11) and (12) involve updating the primal variables of (8) in a one-round Gauss-Seidel fashion; while equations (13), (14) and (15) update the dual variables.

for all $k$ and for all $i,j$. By (16), equations (10) to (15) can be simplified to the following steps

On the other hand, note that the subproblem in (17) is a strongly convex problem. However, it is not easy to handle as subproblem (17) is in fact a min-max (saddle point) problem (see the definition of $\varphi_{i}$ in (7)). Fortunately, by applying the minimax theorem [27, Proposition 2.6.2] and exploiting the strong convexity of (17) with respect to $({\bm{y}}_{i},{\bm{z}}_{i})$, one may avoid solving the min-max problem (17) directly. As we show in Appendix B, $({\bm{y}}_{i}^{k},{\bm{z}}_{i}^{k})$ of subproblem (17) can be conveniently obtained in closed-form as follows

where $({\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k})$ is given by an solution to the following quadratic program (QP)

As also shown in Appendix B, the dummy constraint ${\bm{z}}_{i}={\bm{s}}_{i}$ in (8d) and the augmented term $\frac{\tau_{i}}{2}\!\sum_{i=1}^{N}\|{\bm{z}}_{i}-{\bm{s}}_{i}\|_{2}^{2}$ in (III) are essential for arriving at (22) and (III). Since they are equivalent to applying the proximal minimization method [14, Sec. 3.4.3] to the variables ${\bm{z}}_{i}$’s in (8), we name the developed method above the proximal DC-ADMM method. In Algorithm 1, we summarize the proposed PDC-ADMM method. Note that the PDC-ADMM method in Algorithm 1 is fully parallel and distributed except that, in (29), each agent $i$ requires to exchange ${\bm{y}}_{i}^{k}$ with its neighbors.

The PDC-ADMM method in Algorithm 1 is provably convergent, as stated in the following theorem.

Suppose that Assumptions 1 and 2 hold. Let $({\bm{x}}^{\star},\{{\bm{r}}_{i}^{\star}\}_{i=1}^{N})$ and $({\bm{y}}^{\star},{\bm{z}}^{\star})$, be a pair of optimal primal-dual solution of (5) (i.e., (P)), where ${\bm{x}}^{\star}=[({\bm{x}}_{1}^{\star})^{T},\ldots,({\bm{x}}_{N}^{\star})^{T}]^{T}$ and ${\bm{z}}^{\star}=[({\bm{z}}_{1}^{\star})^{T},\ldots,({\bm{z}}_{N}^{\star})^{T}]^{T}$, and let ${\bm{u}}^{\star}=\{{\bm{u}}_{ij}^{\star}\}$ (which stacks all ${\bm{u}}_{ij}^{\star}$ for all $i,j$) be an optimal dual variable of problem (8). Moreover, let

and $\bar{}{\bm{x}}^{M}=[(\bar{}{\bm{x}}_{1}^{M})^{T},\ldots,(\bar{}{\bm{x}}_{N}^{M})^{T}]^{T}$, where $\{{\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k}\}_{i=1}^{N}$ are generated by (3). Then, it holds that

where $\delta\triangleq\max\{\|{\bm{y}}^{\star}\|_{2},\|{\bm{z}}_{1}^{\star}\|_{2},\ldots,\|{\bm{z}}_{N}^{\star}\|_{2}\},~{}C_{1}\triangleq\frac{\tau_{i}}{2}\max_{\|{\bm{a}}\|_{2}\leq\sqrt{N}}\|{\bm{z}}^{0}-({\bm{z}}^{\star}+{\bm{a}})\|_{\bm{\Gamma}}^{2}+\frac{1}{c}\|{\bm{u}}^{1}-{\bm{u}}^{\star}\|_{2}^{2}+\frac{c}{2}\max_{\|{\bm{a}}\|_{2}\leq 1}\|{\bm{y}}^{0}-{\bf 1}_{N}\otimes({\bm{y}}^{\star}+{\bm{a}})\|^{2}_{{\bm{Q}}}$ and $C_{2}\triangleq\frac{\tau_{i}}{2}\|{\bm{z}}^{0}-{\bm{z}}^{\star}\|_{\bm{\Gamma}}^{2}+\frac{c}{2}\|{\bm{y}}^{0}-{\bf 1}_{N}\otimes{\bm{y}}^{\star}\|^{2}_{{\bm{Q}}}+\frac{1}{c}\|{\bm{u}}^{1}-{\bm{u}}^{\star}\|_{2}^{2}$ are constants, in which ${\bm{\Gamma}}\triangleq{\rm diag}\{\tau_{1},\ldots,\tau_{N}\}$ and ${\bm{Q}}\triangleq({\bm{D}}+{\bm{W}})\otimes{\bm{I}}_{L}\succeq{\bm{0}}$.

The proof is presented in Appendix C. Theorem 1 implies that the proposed PDC-ADMM method asymptotically converges to an optimal solution of (P) with a worst-case $\mathcal{O}(1/k)$ convergence rate.

As discussed in Appendix B, if one removes the dummy constraint ${\bm{z}}_{i}={\bm{s}}_{i}$ from (8) and the augmented term $\!\sum_{i=1}^{N}\frac{\tau_{i}}{2}\|{\bm{z}}_{i}-{\bm{s}}_{i}\|_{2}^{2}$ from (III), then the above development of PDC-ADMM reduces to the existing DC-ADMM method in . The DC-ADMM method is presented in Algorithm 2. Two important remarks on the comparison between PDC-ADMM and DC-ADMM are in order.

As one can see from Algorithm 1 and Algorithm 2, except for the step in (28), the major difference between PDC-ADMM and DC-ADMM lies in (3) and (3). In particular, subproblem (3) is explicitly constrained by the polyhedra constraint ${\bm{C}}_{i}{\bm{x}}_{i}\preceq{\bm{d}}_{i}$; whereas, subproblem (3) has the simple constraint sets ${\bm{x}}_{i}\in\mathcal{S}_{i}$ and ${\bm{r}}_{i}\succeq{\bm{0}}$ only, though (3) has an additional penalty term $\frac{1}{2\tau_{i}}\|{\bm{C}}_{i}{\bm{x}}_{i}+{\bm{r}}_{i}-{\bm{d}}_{i}+\tau_{i}{\bm{z}}_{i}^{k-1}\|_{2}^{2}$. In fact, one can show that, if $\tau_{i}=0$, then the penalty term functions as an indicator function enforcing ${\bm{C}}_{i}{\bm{x}}_{i}+{\bm{r}}_{i}-{\bm{d}}_{i}={\bm{0}}$ (which is equivalent to ${\bm{C}}_{i}{\bm{x}}_{i}\preceq{\bm{d}}_{i}$ as ${\bm{r}}_{i}\succeq{\bm{0}}$). Therefore, (3) boils down to (3) when $\tau_{i}=0$; that is to say, the proposed PDC-ADMM can be regarded as a generalization of DC-ADMM, in the sense that the local polyhedra constraints are handled “softly” depending on the parameter $\tau_{i}$.

On the contrary, since projection onto the polyhedra constraint ${\bm{C}}_{i}{\bm{x}}_{i}\preceq{\bm{d}}_{i}$ has no closed-form and is not trivial to implement in general, previously mentioned algorithms cannot deal with subproblem (3) efficiently. Although primal-dual algorithms (such as ADMM ) can be applied, they are arguably more complex. In particular, since one usually requires a high-accuracy solution to subproblem (3), DC-ADMM is more time consuming than the proposed PDC-ADMM, as will be demonstrated in Section V.

IV Randomized PDC-ADMM

The PDC-ADMM method in Algorithm 1 requires all agents to be active, updating variables and exchanging messages at every iteration $k$. In this section, we develop an randomized PDC-ADMM method which is applicable to networks with randomly ON/OFF agents and non-ideal communication linksThe proposed randomized method and analysis techniques are inspired by the recent works in .. Specifically, assume that, at each iteration (e.g., time epoch), each agent has a probability, say $\alpha_{i}\in(0,1]$, to be ON (active), and moreover, for each link $(i,j)\in\mathcal{E}$, there is a probability $p_{e}\in(0,1]$ to have link failure (i.e., agent $i$ and agent $j$ cannot successfully exchange messages due to, e.g., communication errors). So, the probability that agent $i$ and agent $j$ are both active and able to exchange messages is given by $\beta_{ij}=\alpha_{i}\alpha_{j}(1-p_{e})$. If this happens, we say that link $(i,j)\in\mathcal{E}$ is active at the iteration.

For each iteration $k$, let $\Omega^{k}\subseteq{\mathcal{V}}$ be the set of active agents and let $\Psi^{k}\subseteq\{(i,j)\in\mathcal{E}~{}|i,j\in\Omega^{k}\}$ be the set of active edges. Then, at each iteration $k$ of the proposed randomized PDC-ADMM method, only active agents perform local variable update and they exchange message only with active neighboring agents with active links in between. The proposed randomized PDC-ADMM method is presented in Algorithm 3.

Note that, similar to (18), (19) and (21), update (40) equivalently corresponds to

Besides, if $\Omega^{k}={\mathcal{V}}$ and $\Psi^{k}=\mathcal{E}$ for all $k$, then the randomized PDC-ADMM reduces to the (deterministic) PDC-ADMM in Algorithm 1.

There are two key differences between the randomized PDC-ADMM method and its deterministic counterpart in Algorithm 1. Firstly, in addition to $({\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k},{\bm{y}}_{i}^{k},{\bm{z}}_{i}^{k},{\bm{p}}_{i}^{k})$, each agent $i$ in randomized PDC-ADMM also requires to maintain variables $\{{\bm{t}}_{ij},j\in{\mathcal{N}}_{i}\}$. Secondly, variables $({\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k},{\bm{y}}_{i}^{k},{\bm{z}}_{i}^{k},{\bm{p}}_{i}^{k})$ are updated only if $i\in\Omega^{k}$ and variables $({\bm{t}}_{ij}^{k},\{{\bm{u}}_{ij}^{k},{\bm{v}}_{ji}^{k}\})$ are updated only if $(i,j)\in\Psi^{k}$. Therefore, the randomized PDC-ADMM method is robust against randomly ON/OFF agents and link failures. The convergence result of randomized PDC-ADMM is given by the following theorem.

Suppose that Assumptions 1 and 2 hold. Besides, assume that each agent $i$ has an active probability $\alpha_{i}\in(0,1]$ and, for each link $(i,j)\in\mathcal{E}$, there is a link failure probability $p_{e}\in(0,1]$. Let $({\bm{x}}^{\star},\{{\bm{r}}_{i}^{\star}\}_{i=1}^{N})$ and $({\bm{y}}^{\star},{\bm{z}}^{\star})$, be a pair of optimal primal-dual solution of (5) (i.e., (P)), and let ${\bm{u}}^{\star}=\{{\bm{u}}_{ij}^{\star}\}$ be an optimal dual variable of problem (8). Moreover, let

where $\{{\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k}\}_{i=1}^{N}$ are generated by (3). Then, it holds that

V Numerical Results

The stopping criteria of Algorithms 1 to 3 are based on the solution accuracy ${\sf Acc}=({{\sf obj}({\bm{x}}^{k})-{\sf obj}^{\star}})/{{\sf obj}^{\star}}$ and the feasibility for constraints ${\bm{C}}_{i}{\bm{x}}_{i}\preceq{\bm{d}}_{i}$, $i=1,\ldots,N$, i.e., ${\sf Feas}=\sum_{i=1}^{N}\sum_{j=1}^{P}\max\{({\bm{C}}_{i}{\bm{x}}_{i}^{k}-{\bm{d}}_{i})_{j},0\}/(NP)$, where ${\sf obj}({\bm{x}}^{k})$ denotes the objective value of (2) at ${\bm{x}}^{k}$, and ${\sf obj}^{\star}$ is the optimal value of (2) which was obtained by CVX .

Example 1: We first consider the performance comparison between DC-ADMM and PDC-ADMM. Table I(a) shows the comparison results for $N=50$, $K=500$, $L=100$, $P=250$ and $\lambda=10$. For PDC-ADMM, we simply set $\tau_{1}=\cdots=\tau_{N}\triangleq\tau$ and $\tau=c$. The penalty parameters $c$ of the two algorithms are respectively chosen so that the two algorithms can exhibit best convergence behaviorsWe did not perform exhaustive search. Instead, we simply pick the value of $c$ from the set $\{0.0005,0.001,0.005,0.1,0.5,1,5,10,50,100\}$ for which the algorithm can yield best convergence behavior for a randomly generated problem instance and graph. Once the value of $c$ is determined, it is fixed and tested for another 9 randomly generated problem instances and graphs.. One can see from Table I(a) that DC-ADMM ($c=0.01$, $c_{1}=5$, $\epsilon_{1}=10^{-6}$)The parameter $c_{1}$ is also chosen in a similar fashion as the parameter $c$. can achieve the stopping condition Acc $+$ Feas $\leq 10^{-4}$ with an average iteration number $37.7$ but spends an average per-agent computation time of 19.63 seconds. One should note that a naive way to reducing the computation time of DC-ADMM is to reduce the solution accuracy of subproblem (3), i.e., increasing $\epsilon_{1}$. As seen, DC-ADMM with $\epsilon_{1}=10^{-5}$ has a reduced per-agent computation time 9.87 seconds; however, the required iteration number drastically increases to 980.1. By contrast, one can see from Table I(a) that the proposed PDC-ADMM ($c=\tau=0.01$, $\epsilon_{2}=10^{-6}$) can achieve the stopping condition with an average iteration number $55.9$ and a much less (per-agent) computation time $5.76$ seconds. If one reduces the solution accuracy of BSUM for solving subproblem (3) to $\epsilon_{2}=10^{-5}$, then the computation time of PDC-ADMM can further reduce to 1.58 seconds, though the required iteration number is increased to 298.8. Figure 1 displays the convergence curves of DC-ADMM and PDC-ADMM for one of the 10 randomly generated problem instances. One can see from Fig. 1(a) that PDC-ADMM ($c=\tau=0.01$, $\epsilon_{2}=10^{-6}$) has a comparable convergence behavior as DC-ADMM ($c=0.01$, $c_{1}=5$, $\epsilon_{1}=10^{-6}$) when with respect to the iteration number and in terms of the solution accuracy Acc. Moreover, as seen from Fig. 1(b), when with respect to the computation time, PDC-ADMM ($c=\tau=0.01$, $\epsilon_{2}=10^{-6}$) is much faster than DC-ADMM ($c=0.01$, $c_{1}=5$, $\epsilon_{1}=10^{-6}$). However, it is seen from Fig. 1(c) that DC-ADMM usually has a small value of feasibility Feas which is understandable as the constraint ${\bm{C}}_{i}{\bm{x}}_{i}\preceq{\bm{d}}_{i}$ is explicitly handled in subproblem (3); whereas the constraint feasibility associated with PDC-ADMM gradually decreases with the iteration number. This explains why in Table I(a), to achieve Acc $+$ Feas $\leq 10^{-4}$, PDC-ADMM always has smaller values of Acc than DC-ADMM but has larger values of Feas.

In summary, by comparing to the naive strategy of reducing the solution accuracy of (3) in DC-ADMM, we observe that the proposed PDC-ADMM can achieve a much better tradeoff between the iteration number and computation time. Since the iteration number is also the number of message exchanges between connecting agents, the results equivalently show that the proposed PDC-ADMM achieves a better tradeoff between communication overhead and computational complexity.

In Table I(b), we present another set of simulation results for $N=50$, $K=1,000$, $L=100$, $P=500$ and $\lambda=100$. One can still observe that the proposed PDC-ADMM has a better tradeoff between the iteration number and computation time compared to DC-ADMM. In particular, one can see that, for DC-ADMM with $\epsilon_{1}$ reduced from $\epsilon_{1}=10^{-6}$ to $\epsilon_{1}=10^{-5}$, reduction of the computation time is limited but the iteration number increased to a large number of 1173.

Example 2: In this example, we examine the convergence behavior of randomized PDC-ADMM (Algorithm 3). It is set that $\alpha\triangleq\alpha_{1}=\cdots=\alpha_{N}$, i.e., all agents have the same active probability. Note that, for $\alpha=1$ and $p_{e}=0$, randomized PDC-ADMM performs identically as the PDC-ADMM in Algorithm 1. Figure 2 presents the convergence curves of randomized PDC-ADMM for different values of $\alpha$ and $p_{e}$ and for the stopping condition being Acc $+$ Feas $\leq 10^{-4}$. The simulation setting and problem instance are the same as that used for PDC-ADMM ($c=\tau=0.05$, $\epsilon_{2}=10^{-5}$) in Fig. 1. One can see from Fig. 2(a) that no matter when $\alpha$ decreases to $0.7$ and/or $p_{e}$ increases to $0.5$, randomized PDC-ADMM always exhibits consistent convergence behavior, though the convergence speed decreases accordingly. We also observe from Fig. 2(a) that Acc may oscillate in the first few iterations when $\alpha<1$ and $p_{e}>0$. Interestingly, from Fig. 2(b), one can observe that the values of $\alpha$ and $p_{e}$ do not affect the convergence behavior of constraint feasibility much.

VI Conclusions

In this paper, we have proposed two ADMM based distributed optimization methods, namely the PDC-ADMM method (Algorithm 1) and the randomized PDC-ADMM method (Algorithm 3) for the polyhedra constrained problem (P). In contrast to the existing DC-ADMM where each agent requires to solve a polyhedra constrained subproblem at each iteration, agents in the proposed PDC-ADMM and randomized PDC-ADMM methods deal with a subproblem with simple constraints only, thereby more efficiently implementable than DC-ADMM. For both proposed PDC-ADMM and randomized PDC-ADMM, we have shown that they have a worst-case $\mathcal{O}(1/k)$ convergence rate. The presented simulation results based on the constrained LASSO problem in (2) have shown that the proposed PDC-ADMM method exhibits a much lower computation time than DC-ADMM, although the required iteration number is larger. It has been observed that the tradeoff between communication overhead and computational complexity of PDC-ADMM is much better, especially when comparing to the naive strategy of reducing the subproblem solution accuracy of DC-ADMM. It has been also shown that the proposed randomized PDC-ADMM method can converge consistently in the presence of randomly ON/OFF agents and severely unreliable links.

VII Acknowledgement

The author would like to thank Prof. Min Tao at the Nanjing University for her valuable discussions.

Appendix A Proof of Equation (16)

It is easy to derive from (10) and (11) that ${\bm{t}}_{ij}^{k}$ and ${\bm{s}}_{i}^{k}$ have close-form solutions as

respectively. By substituting (A.1) into (14) and (15), respectively, followed by summing the two equations, one obtains

On the other hand, it directly follows from (A.2) and (13) that ${\bm{w}}_{i}^{k}={\bm{0}}$ and ${\bm{s}}_{i}^{k}={\bm{z}}_{i}^{k}~{}\forall i,k.$ $\blacksquare$

Appendix B Proof of Equations (22) and (III)

By (7) and (20), subproblem (17) can be explicitly written as a min-max problem as follows

Notice that $\hat{}{\mathcal{L}}({\bm{x}}_{i},{\bm{r}}_{i},{\bm{y}}_{i},{\bm{z}}_{i})$ is (strongly) convex with respect to $({\bm{y}}_{i},{\bm{z}}_{i})$ given any $({\bm{x}}_{i},{\bm{r}}_{i})$ and is concave with respect to $({\bm{x}}_{i},{\bm{r}}_{i})$ given any $({\bm{y}}_{i},{\bm{z}}_{i})$. Therefore, the minimax theorem [27, Proposition 2.6.2] can be applied so that saddle point exists for (B) and it is equivalent to its max-min counterpart

Let $({\bm{y}}_{i}^{k},{\bm{z}}_{i}^{k})$ and $({\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k})$ be a pair of saddle point of (B) and (A.7). Then, given $({\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k})$, $({\bm{y}}_{i}^{k},{\bm{z}}_{i}^{k})$ is the unique inner minimizer of (A.7), which, from (B), can be readily obtained as the closed-form solutions in (22) and (22b), respectively. By substituting (22) and (22b) into (A.7), $({\bm{x}}_{i}^{k},{\bm{r}}_{i}^{k})$ can be obtain by subproblem (III).

We remark here that if one removes the dummy constraint ${\bm{z}}_{i}={\bm{s}}_{i}$ from (8) and the augmented term $\!\sum_{i=1}^{N}\frac{\tau_{i}}{2}\|{\bm{z}}_{i}-{\bm{s}}_{i}\|_{2}^{2}$ from (III), then the corresponding subproblem (17) reduces to

Note that (B) is no longer strongly convex with respect to ${\bm{z}}_{i}$ as the term $\frac{\tau_{i}}{2}\!\|{\bm{z}}_{i}-{\bm{z}}_{i}^{k-1}\|_{2}^{2}$ is absent. After applying the minmax theorem to (B):

one can see that, to have a bounded optimal value for the inner minimization problem, it must hold ${\bm{C}}_{i}{\bm{x}}_{i}+{\bm{r}}_{i}-{\bm{d}}_{i}={\bm{0}}$, and thus ${\bm{z}}_{i}$ appears redundant. Moreover, one can show that the inner optimal ${\bm{y}}$ is

The variable ${\bm{r}}_{i}$ also appears redundant and can be removed from (B). The resultant steps of (B), (B) and (21) are the DC-ADMM method in (see Algorithm 2). $\blacksquare$

Appendix C Proof of Theorem 1

Let us equivalently write (3) to (29) as follows: $\forall i\in{\mathcal{V}},$

Notice that we have recovered $\{{\bm{u}}_{ij}^{k-1},{\bm{v}}_{ji}^{k-1}\}$ from ${\bm{p}}_{i}^{k-1}$ according to (18), (19), and (21). Besides, the update orders of $({\bm{u}}_{ij},{\bm{v}}_{ji})$ and $({\bm{x}}_{i},{\bm{r}}_{i},{\bm{y}}_{i},{\bm{z}}_{i})$ are reversed here for ease of the analysis.

According to [14, Lemma 4.1], the optimality condition of (C) with respect to ${\bm{x}}_{i}$ is given by: $\forall{\bm{x}}_{i}\in\mathcal{S}_{i}$,

where the equality is obtained by using (A.14) and (A.15). Analogously, the optimality condition of (C) with respect to ${\bm{r}}_{i}$ is given by, $\forall{\bm{r}}_{i}\succeq{\bm{0}}$,

where the equality is owing to (A.15). By summing (C) and (C), one obtains

By letting ${\bm{x}}_{i}={\bm{x}}_{i}^{\star}$ and ${\bm{r}}_{i}={\bm{r}}_{i}^{\star}$ for all $i\in{\mathcal{V}}$ in (C), where $({\bm{x}}_{i}^{\star},{\bm{r}}_{i}^{\star})_{i=1}^{N}$ denotes the optimal solution to problem (5), we have the following chain from (C)

where the first equality is due to the fact ${\bm{C}}_{i}{\bm{x}}_{i}^{\star}+{\bm{r}}_{i}^{\star}={\bm{d}}_{i}$; the second equality is obtained by adding and subtracting both terms ${\bm{y}}^{T}{\bf E}_{i}({\bm{x}}_{i}^{k}-{\bm{x}}_{i}^{\star})$ and ${\bm{z}}_{i}^{T}({\bm{C}}_{i}{\bm{x}}_{i}^{k}+{\bm{r}}_{i}^{k}-{\bm{d}}_{i})$ for arbitrary ${\bm{y}}$ and ${\bm{z}}_{i}$; the last equality is due to (A.15).

On the other hand, note that (A.14) can be expressed as

where the last equality is obtained by applying (A.11) and (A.12). Furthermore, let $({\bm{y}}_{i}^{\star},{\bm{z}}_{i}^{\star})_{i=1}^{N}$ be an optimal solution to problem (8), and denote $(\{{\bm{u}}_{ij}^{\star}\},\{{\bm{v}}_{ij}^{\star}\})$ be an optimal dual solution of (8). Then, according to the Karush-Kuhn-Tucker (KKT) condition , we have

where $\partial_{{\bm{y}}_{i}}\varphi({\bm{y}}_{i}^{\star},{\bm{z}}_{i}^{\star})$ denotes a subgradient of $\varphi$ with respect to ${\bm{y}}_{i}$ at point $({\bm{y}}_{i}^{\star},{\bm{z}}_{i}^{\star})$. Since ${\bm{y}}^{\star}\triangleq{\bm{y}}_{i}^{\star}=\cdots={\bm{y}}_{N}^{\star}$ under Assumption 1, $({\bm{x}}_{i}^{\star},{\bm{r}}_{i}^{\star})_{i=1}^{N}$ and $({\bm{y}}^{\star},\{{\bm{z}}_{i}^{\star}\}_{i=1}^{N})$ form a pair of primal-dual solution to problem (5) under Assumption 2, $({\bm{x}}_{i}^{\star},{\bm{r}}_{i}^{\star})$ is optimal to (7) given $({\bm{y}},{\bm{z}}_{i})=({\bm{y}}_{i}^{\star},{\bm{z}}_{i}^{\star})$, and thus $\partial_{{\bm{y}}_{i}}\varphi({\bm{y}}_{i}^{\star},{\bm{z}}_{i}^{\star})=-{\bf E}_{i}{\bm{x}}_{i}^{\star}$ , which and (A.22) give rise to

By combing (C) and (A.23) followed by multiplying $({\bm{y}}_{i}^{k}-{\bm{y}})$ on both sides of the resultant equation, one obtains

By further substituting (C) into (C) and summing for $i=1,\ldots,N$, one obtains that

for arbitrary ${\bm{y}}$ and ${\bm{z}}_{1},\ldots,{\bm{z}}_{N}$, where ${\bm{x}}^{k}=[({\bm{x}}_{1}^{k})^{T},\ldots,({\bm{x}}_{N}^{k})^{T}]^{T}$.

By following the same idea as in [19, Eqn. (A.16)], one can show that

where ${\bm{y}}^{k}=[({\bm{y}}_{1}^{k})^{T},\ldots,({\bm{y}}_{N}^{k})^{T}]^{T}$, $\hat{}{\bm{y}}\triangleq{\bf 1}_{N}\otimes{\bm{y}}$ and ${\bm{Q}}\triangleq({\bm{D}}+{\bm{W}})\otimes{\bm{I}}_{L}\succeq{\bm{0}}$ (see [19, Remark 1]). Moreover, according to [19, Eqn. (A.15)], it can be shown that

where ${\bm{u}}^{k}$ (${\bm{u}}^{\star}$) is a vector that stacks ${\bm{u}}_{ij}^{k}$ (${\bm{u}}_{ij}^{\star}$) for all $j\in\mathcal{N}_{i}$ and $i\in{\mathcal{V}}$. As a result, (C) can be expressed as

where ${\bm{z}}^{k}=[({\bm{z}}_{1}^{k})^{T},\ldots,({\bm{z}}_{N}^{k})^{T}]^{T}$, ${\bm{z}}=[{\bm{z}}_{1}^{T},\ldots,{\bm{z}}_{N}^{T}]^{T}$ and ${\bm{\Gamma}}\triangleq{\rm diag}\{\tau_{1},\ldots,\tau_{N}\}$. By applying the fact of

for any sequence ${\bm{a}}^{k}$ and matrix ${\bm{A}}\succeq{\bm{0}}$, to (C), we obtain

Summing (C) for $k=1,\ldots,M,$ and taking the average gives rise to

where $\bar{}{\bm{x}}_{i}^{M}\triangleq\frac{1}{M}\sum_{k=1}^{M}{\bm{x}}_{i}^{k},~{}\bar{}{\bm{r}}_{i}^{M}\triangleq\frac{1}{M}\sum_{k=1}^{M}{\bm{r}}_{i}^{k},$ and the last inequality is owing to the convexity of $F$ (Assumption 2).

Let ${\bm{y}}={\bm{y}}^{\star}+\frac{\sum_{i=1}^{N}{\bf E}_{i}\bar{}{\bm{x}}_{i}^{M}-{\bm{q}}}{\|\sum_{i=1}^{N}{\bf E}_{i}\bar{}{\bm{x}}_{i}^{M}-{\bm{q}}\|_{2}}$ and ${\bm{z}}_{i}={\bm{z}}_{i}^{\star}+\frac{{\bm{C}}_{i}\bar{}{\bm{x}}_{i}^{M}+\bar{}{\bm{r}}_{i}^{M}-{\bm{d}}_{i}}{\|{\bm{C}}_{i}\bar{}{\bm{x}}_{i}^{M}+\bar{}{\bm{r}}_{i}^{M}-{\bm{d}}_{i}\|_{2}}$ $\forall i\in{\mathcal{V}}$, in (C). Moreover, note that

according to the duality theory . Thus, we obtain that

On the other hand, let ${\bm{y}}={\bm{y}}^{\star}$ and ${\bm{z}}_{i}={\bm{z}}_{i}^{\star}$ $\forall i\in{\mathcal{V}}$, in (C). Then, we have that

where $\delta\triangleq\max\{\|{\bm{y}}^{\star}\|_{2},\|{\bm{z}}_{1}^{\star}\|_{2},\ldots,\|{\bm{z}}_{N}^{\star}\|_{2}\}$. Using (C), (C) implies that

where $C_{2}\triangleq\frac{1}{2}\|{\bm{z}}^{0}-{\bm{z}}^{\star}\|_{\bm{\Gamma}}^{2}+\frac{c}{2}\|{\bm{y}}^{0}-{\bf 1}_{N}\otimes{\bm{y}}^{\star}\|^{2}_{{\bm{Q}}}+\frac{1}{c}\|{\bm{u}}^{1}-{\bm{u}}^{\star}\|_{2}^{2}$. After summing (C) and (A.35), one obtains (49). $\blacksquare$

Appendix D Proof of Theorem 2

The proof is based on the “full iterates” assuming that all agents and all edges are active at iteration $k$. Specifically, the full iterates for iteration $k$ are

Let us consider the optimality condition of (D). Following similar steps as in (C) to (C), one can have that

Besides, note that (D) can be expressed as

By substituting (D) into (D) and summing the equations for $i=1,\ldots,N$, one obtains

where the first equality is obtained by the fact that, for any $\{\alpha_{ij}\}$,

Then, similar to the derivations from (C) to (C), one can deduce from (D) and (D) that

To connect the full iterates with the instantaneous iterates, let us define a weighed Lagrangian as

where the last inequality is due to (D). Furthermore, define

Then, by (A.42), (A.43) and (3), one can show that

By substituting (D), (A.58) and (A.59) into (D) followed by taking the expectation with respect to ${\mathcal{J}}_{k-1}$, one obtains

Upon summing the above equation from $k=1,\ldots,M$, and taking the average, we can obtain the following bound

Also similar to (C) and (A.35), by letting ${\bm{y}}={\bm{y}}^{\star}$ and ${\bm{z}}={\bm{z}}^{\star}$, one can bound the expected objective value as

where $\delta\triangleq\max\{\|{\bm{y}}^{\star}\|_{2},\|{\bm{z}}_{1}^{\star}\|_{2},\ldots,\|{\bm{z}}_{N}^{\star}\|_{2}\}$ and

The proof is complete by adding (D) and (A.63). $\blacksquare$

References