In recent years, the communication, signal processing, control, and optimization communities have witnessed considerable research efforts on decentralized optimization for networked multi-agent systems [1–3]. A networked multi-agent system, such as a wireless sensor network (WSN) or a networked control system (NCS), is composed of multiple geographically distributed but interconnected agents which have sensing, computation, communication, and actuating abilities. This system generally has limited resources for communication, since battery power is limited and recharging is difficult, while communication between two agents is energy-consuming. Furthermore, the communication link is often vulnerable and bandwidth-limited. In this situation, decentralized optimization emerges as an effective approach to improve network scalability. In decentralized optimization, data and computation are decentralized. Each agent exchanges information with its neighbors and accomplishes an otherwise centralized optimization task.

This article focuses on the

*decentralized consensus optimization* problem. We consider a network of

*L* agents which cooperatively optimize a separable objective function [

3–

8]:

$min\phantom{\rule{1em}{0ex}}\sum _{i=1}^{L}{f}_{i}\left(x\right),$

(1)

where ${f}_{i}\left(x\right):{\mathcal{R}}^{N}\to \mathcal{R}$ is a convex function known to agent *i* only. The goal is to minimize the objective subject to consensus on *x*.

### Related study

The decentralized consensus optimization formulation (1) arises in many practical applications, such as averaging [9–11], estimation [12–17], learning [18–21], etc. The form of *f*_{
i
}(*x*) can be least squares [11–13], *ℓ*_{1}-regularized least squares [14–17], or more general ones [18–21]. Note that this model can be extended to account for those with separable constraints, such as the network utility maximization (NUM) problem [22–24].

Existing approaches to solving (1) include: i) belief propagation based on graphical models and Markovian random fields [18–20]; ii) incremental optimization which minimizes the overall objective function along a predefined path on the network [7, 8]; iii) stochastic optimization with information exchange between neighboring agents [4–6]; and iv) optimization with explicit consensus constraints which can be handled with the alternating direction method (ADM) [3, 12–17]. The ADM approach is fully decentralized, does not make any assumptions on network infrastructure such as free of loop or with a predefined path, and generally has satisfactory convergence performance. In this article, we mainly discuss the application of ADMs in the decentralized consensus optimization problem.

Our research is along the line of information-driven signal processing and control of WSNs and NCSs [24–26]. Accompanied with the unprecedented data collection abilities offered by large-scale networked multi-agent systems, a new challenge also arises: *how should we process such a large amount of data to make estimates and produce control strategies given limited network resources?* Instead of processing the data in a fusion center, our solution is letting each agent autonomously make decisions aided by limited communication with its neighbors. From this perspective, each individual objective function *f*_{
i
}(*x*) in (1) is constructed from the data collected by agent *i*, and *x* is the global information common to all agents (e.g., estimates or control strategies) obtained based on the data collected by the whole network. Though this framework can be generalized to various signal processing and control problems, this article focuses on those can be formulated as (1). For problems such as dynamic control and Kalman filtering of networked multi-agent systems, interested readers are referred to [1, 2, 27, 28], respectively.

### Our contribution

Motivated by a series of recent articles on multi-block ADMs and their convergence analysis [29–31], this article describes their applications to the decentralized consensus optimization problem. The multi-block ADM with parallel spliting is reviewed in Section 3. Unlike the classical ADM (see textbooks [32, 33]), this multi-block ADM splits the optimization variables into *multiple* blocks and sequentially updates just one of them while fixing the others. The classical ADM, on the other hand, only has two blocks of variables. Hence in this article we refer to it by the two-block ADM. Our problem (1) does not naturally have two distinct blocks of variables, and to apply the two-block ADM one needs to introduce extra variables (see e.g., [15, 16, 32]). We review this in Section 2. On the other hand, it is simpler to apply the multi-block ADM to (1) and the resulting algorithm is readily decentralized.

In this article also analyzes the convergence rate of the multi-block ADM applied to the average consensus problem, which is a special case of (1) where ${f}_{i}\left(x\right)=\frac{1}{2}\parallel x-{b}_{i}{\parallel}_{2}^{2}$ for all *i*. In this setting, if the parameters of the multi-block ADM satisfy a certain formula, it is equivalent to the two-block ADM. Therefore, the two-block ADM can be considered as a special case of the multi-block ADM on average consensus problems. This relation also gives a guideline to select the parameters of the multi-block ADM so that it is not equivalent to and runs faster than the two-block ADM on all the tested decentralized consensus optimization problems, including the tested average consensus problems. The simulation results demonstrate that the multi-block ADM accelerates convergence, reduces communication cost, and thus improves network scalability.

### Paper organization

The rest of this article is organized as follows. Section 2 reviews a reformulation of the decentralized consensus optimization problem (1), to which the two-block ADM is applied. Section 3 reviews the multi-block ADM and applies a parallel-splitting version of it to (1). Section 4 elaborates on the convergence rate analysis on the average consensus problem, and shows that the two-block ADM is a special case of the multi-block ADM in this case. Section 5 presents numerical simulations of the two-block and multi-block ADMs. Finally, Section 6 concludes the article. Appendix Appendix 1 is placed in the last section.