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This paper focuses on a class of event-triggered discrete-time distributed consensus optimization algorithms, with a set of agents whose communication topology is depicted by a sequence of time-varying networks. The communication process is steered by independent trigger conditions observed by agents and is decentralized and just rests with each agent’s own state. At each time, each agent only has access to its privately local Lipschitz convex objective function. At the next time step, every agent updates its state by applying its own objective function and the information sent from its neighboring agents. Under the assumption that the network topology is uniformly strongly connected and weight-balanced, the novel event-triggered distributed subgradient algorithm is capable of steering the whole network of agents asymptotically converging to an optimal solution of the convex optimization problem. Finally, a simulation example is given to validate effectiveness of the introduced algorithm and demonstrate feasibility of the theoretical analysis.

In the last decade, multiagent systems have obtained some achievements in theory and application, like consensus problem [

Among the existing papers, the consensus-based subgradient methods for solving the distributed convex optimization problem have drawn a surge of attentions since Nedic and Ozdaglar presented a systematic analysis of it in [

Our method builds on the pioneering work of [

The remainder of this paper is organized as follows. Some essential concepts and knowledge with regard to graph theory are given, and problems are formulated in Section

In this section, we present some important mathematical preliminaries including algebraic graph theory, notations, and problem formulation (referring to [

We always employ a graph to describe the information exchange between the nodes. The information exchange between

Some mathematically standard notations throughout this paper are listed in the following.

In this subsection, we consider a network of

In this paper, we do not assume the differentiability of the local objective function

The following assumptions are necessary in the analysis of distributed optimization algorithm throughout this paper.

Let

The subgradients

In this section, motivated by [

Consider a set

Denote the measurement error as

Before giving some supporting lemmas, we need the following assumption on the sequences of step-sizes and

The step-sizes

There exists a constant

Next, the transition matrices are introduced as follows:

Let the weight-balanced Assumption

We refer the reader to the papers [

Before moving on, it is important to introduce the following lemma and the proof of the lemma.

Let

We do not give the proof of Lemma

Chen and Ren [

We now define the triggering time sequence

Since

Lemma

We next introduce the characteristic that the agents reach a consensus asymptotically, which means the agent estimates

Let the weight-balanced Assumption

It is obtained from (

We now begin to introduce a well-known convergence result which is shown in the following lemma.

Let

The proof process can imitate that of [

In what follows, we present a key lemma, which is important in the analysis of the distributed optimization algorithm. Thereafter, we study the convergence behavior of the subgradient algorithm, where the optimal solution can be asymptotically reached.

Consider an optimization problem

The proof procedure can imitate that of Lemma 7 in [

Let the weight-balanced Assumption

Applying (

In this section, a numerical example is presented to verify the feasibility of the proposed algorithm and correctness of our theoretical analysis. Consider the time-varying communication graph

Consider the optimization problem (

All agents’ states

Evolution of all agents’ control inputs

All agents’ sampling time instant sequences

Evolution of measurement error and threshold for agent

In this paper, a novel consensus-based event-triggered algorithm for solving the distributed convex optimization problem over time-varying directed networks has been analyzed in detail. We have proved that, based on the designed distributed event-triggered scheme and the uniformly strongly connected communication graph sequence

The authors declare that they have no conflicts of interest.

This work described in this paper was supported in part by the Visiting Scholarship of State Key Laboratory of Power Transmission Equipment & System Security and New Technology (Chongqing University) under Grant 2007DA10512716421, in part by the Fundamental Research Funds for the Central Universities under Grant XDJK2016B016, in part by the Natural Science Foundation Project of Chongqing CSTC under Grant cstc2014jcyjA40016, in part by the China Postdoctoral Science Foundation under Grant 2016M590852, and in part by the Natural Science Foundation of China under Grant 61403314.