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In this paper, the edge-based and node-based adaptive algorithms are established, respectively, to solve the distribution convex optimization problem. The algorithms are based on multiagent systems with general linear dynamics; each agent uses only local information and cooperatively reaches the minimizer. Compared with existing results, a damping term in the adaptive law is introduced for the adaptive algorithms, which makes the algorithms more robust. Under some sufficient conditions, all agents asymptotically converge to the consensus value which minimizes the cost function. An example is provided for the effectiveness of the proposed algorithms.

Recently, the consensus problems of multiagent systems have been extensively investigated on account of its widespread application, such as cooperative reconnaissance and unmanned aerial vehicles formation. Many meaningful results about consensus algorithms [

Meanwhile, with the development of various network systems, various distributed optimization problems have emerged. Extensive efforts to design the efficient algorithm have been put into the distributed optimization problems. Because of the advantages of decentralized and distributed structure, the distributed algorithms based on multiagent systems are more efficient than the centralized algorithm. A large number of consensus-based optimization algorithms have been presented over the past two decades. For instance, in [

In the above works, the proposed optimization algorithms depend on the selection of constant control gains. In fact, the lower bound of the gains is determined by the smallest nonzero eigenvalue of Laplacian matrix for the communication graph, i.e., algebraic connectivity which is global information. These consensus algorithms are not fully distributed. To face this challenge, some well-performing adaptive algorithms have been proposed, where the adaptive gain updating laws rely on local information of the agents. For example, in [

Note that more agent networks in practical applications are described by generalized linear systems. Moreover, from the results in [

The article is organized as follows. In next section, some preliminaries and lemmas are introduced. In Section

In this paper,

An undirected graph

The undirected graph

Under Assumption

Let

If

Consider the following multiagent system; each agent satisfies the dynamics:

Our aim is to design

We give the following hypothesis.

The local cost function

The equivalent characterization for (

Each set

To solve problem (

From (

Note that

Under Assumptions

Choose the Lyapunov function candidate

Take

Let consensus value

Let

Here, damping terms

To solve problem (

From (

Denote

Under Assumptions

Choose the Lyapunov function candidate

Calculate the derivative of

It is easy to see that the node-based algorithm is very different from the edge-based algorithm. The advantage is that the nosed-based algorithm has a less computation and more concise form than the edge-based algorithm. The disadvantage is that the nosed-based algorithm converges more slowly than the edge-based algorithm.

Based on the edge and node standpoints, two adaptive algorithms have been established for solving the distributed convex optimization problems; the algorithms are fully distributed without depending on any global information; that is, the adaptive algorithms can get the optimal solution via local information. Moreover, the connectivity of the networks takes a key role in the rate of convergence of the algorithms during the calculation process.

Consider the following distributed optimization:

The communication topology for the distributed optimization.

The state variables

The adaptive law under the edge-based algorithm.

The global cost function under the edge-based algorithm.

The state variables

The adaptive law under the node-based algorithm.

The global cost function under the node-based algorithm.

In this paper, two distributed adaptive algorithms to solve the distributed convex optimization problems are designed based on general linear multiagent systems from the edge-based and node-based standpoint. Compared with the existing algorithms, through introducing a damping term in the update law, the algorithms are robust in the face of input disturbances. In this paper, the multiagent networks are undirected graph. Next, we will focus on the case that the multiagent networks are directed graph.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This work was supported by the Scientific Research Program of the Higher Education Institution of Xinjiang (Grant no. XJEDU2017T001), the Natural Science Foundation of Xinjiang Uygur Autonomous Region (Grant no. 2018D01C039), and the National People’s Republic of China (Grants nos. U1703262 and 61563048).