Abstract
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.
1. Introduction
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 [1–7] have been established. Specially, the consensus problems can solve the distributed optimization, when the consensus value minimizes the global cost function, which is a sum of local objective functions each of which is known by one agent.
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 [8, 9], the subgradient algorithms upon multiagent systems have been presented. In [10], the algorithms with fixed step size have been proposed. In [11], edge-based fixed-time consensus algorithms have been established. More results can be found in [12–15].
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 [16, 17], the adaptive algorithms of the finite time convergence have been established upon integrator multiagent systems. In [18], the adaptive algorithms have been proposed upon general linear multiagent systems from the edge-based and node-based design.
Note that more agent networks in practical applications are described by generalized linear systems. Moreover, from the results in [19], the control gains in [17, 18] monotonically increase very quickly when the value of the initial states is large. Overlarge control gains will magnify the control input and will lead to unstable algorithms. Based on the above considerations, from the edge-based and node-based standpoint two distributed adaptive algorithms for solving the distributed convex optimization problems are proposed upon general linear multiagent systems in this paper. Compared with the existing algorithms, the proposed algorithms based on generalized linear dynamical systems are more practical and its theoretical analysis is more difficult than that based on integrator systems. The algorithm does not require global information, which makes it fully distributed and easier to implement. The proposed algorithms introduce a damping term in the update law, which makes the algorithm robust under input disturbance and the high-gain update law.
The article is organized as follows. In next section, some preliminaries and lemmas are introduced. In Section 3, the edge-based and node-based consensus protocols are designed for solving the distributed convex optimization; the asymptotic convergence is guaranteed for the adaptive algorithms. In Section 4, the effectiveness of the performance for the algorithm to solve the distributed convex optimization problems is presented by a typical example. In Section 5, a brief conclusion is given.
2. Notations and Preliminaries
In this paper, is the set of all -dimensional real vectors. is the set of real matrices. means the vector that all the elements are one. is the -dimensional identity matrix. For a real matrix , means is positive definite (semidefinite). represents the transpose. , , and denote the 1-norm, 2-norm and infinite norm, respectively. Given , represents the gradient of the function . The function is , if and if for any element for ,
An undirected graph consists of a node set and an edge set , the edge if and only if , and the self-edge is not allowed. The incidence matrix corresponds to a graph with an arbitrary orientation, namely, whose edges have a terminal node and initial node. For th edge, if the node is initial, , if the node is terminal, ; otherwise, . The graph Laplacian of is defined as . A path in the graph G from to is a sequence of distinct nodes starting with and ending with . The graph G is connected, if there is a path between any two nodes.
Assumption 1. The undirected graph is connected.
Lemma 2 (see [20]). Under Assumption 1, is positive semidefinite, is a simple eigenvalue for , and is the associated eigenvector; its smallest nonzero eigenvalue .
Lemma 3 (see [21]). Let be a continuously differentiable convex function; then is minimized if and only if .
Lemma 4 (see [22]). If , and , then .
3. Problem Formulation
Consider the following multiagent system; each agent satisfies the dynamics:where is the protocol, is the state for th agent, and and are the constant matrices.
Our aim is to design for (1) by using only local information, such that all agents cooperatively arrive the consensus value which minimizes the convex problemwhere is a local cost function, which is known only to agent .
We give the following hypothesis.
Assumption 5. The local cost function is convex and differentiable; its gradient satisfieswhere are defined in (1), , is some negative define matrix, is continuous, and there exists such that , , for all .
The equivalent characterization for (2) is that the agents achieve consensus value which minimizes the function , namely,
Assumption 6. Each set is nonempty; that is to say, there exists such that is minimum.
4. Edge-Based Adaptive Designs
To solve problem (3), the edge-based adaptive algorithm for (1) is presented as follows:where are adaptive coupling strengths, is a feedback matrix, the initial states satisfy , and .
From (5), system (1) can be written as follows:Denote and . Thus, (9) can be represented by the compact formwhere is the incidence matrix, , and . The solution of the system (10) is understood by the sense of Filippov [23].
Note that ; multiplying both sides of (10) by , we get the error systemwhere , . Obviously, if and only if .
Theorem 7. Under Assumptions 1–6 and Lemma 3, problem (4) can be solved by algorithm (5), if there exists the feedback matrix , such thatwhere is the positive-definite matrix. Meanwhile, as , we have and will converge to the same positive number.
Proof. Choose the Lyapunov function candidatewhere and are the positive constants; is the positive-definite matrix.
Take ; we haveFrom the third term in (14) and , we getAccording to the fourth term in (14), we obtain Note thatby the fourth term in (14), we obtain From (15), (16), and (19), we getand choose , , due to (12),i.e., is nonincreasing. Moreover, from (13), we have . Meanwhile, integrating both sides of (21), one has . Thus, . From Lemma 4, ; that is,
Let consensus value ; then According to Assumption 5, we havethus,From Assumption 1, is bounded below; that is, . Thus, we obtain . From Lemma 2, when , minimizes the cost function .
Let , thenTo solve (25), one hasBy , we have . From , we obtain . By (26), . Now, let us prove . Consider the following two cases.
is bounded. Then, Then, Based on and is monotonically increasing, when , we acquire that will converge to some positive constant. From , we have and will converge to the same positive number. For and , we have the same results.
Remark 8. Here, damping terms are designed for the gain adaption law of the adaptive algorithm (5). increases monotonically and converges to a finite positive constant. Choosing , by (26), , and , we have that will increase from zero and then decrease to zero. In other words, will increase at first and then decrease for some time upon the select of and . For , we have similar results. Compared with the adaptive law in [14, 15], the advantage of the adaptive law introducing the damping term is that the adaptive control gains and the amplitude of the control inputs are smaller; this makes our algorithm more robust.
5. Node-Based Adaptive Designs
To solve problem (3), the node-based protocol for (1) is proposed as follows:where are adaptive coupling strengths, is a feedback matrix, and the initial states satisfy .
From (1) and (29), the closed-loop system can be represented as follows:where the first and third terms take the role to minimize the function , and the second term makes the agents achieve consensus.
Denote , , and ; by (32) we obtainNote that ; multiply both sides of (33) by ; the following error system is obtained:where , .
Theorem 9. Under Assumptions 1–6 and Lemma 3, problem (29) can be solved by the nosed-based algorithm (5) if , where satisfiesMeanwhile, as , we have and will converge to the same positive number.
Proof. Choose the Lyapunov function candidatewhere ; is a positive constant.
Calculate the derivative of along system (34) and choose ; we haveFrom the third term in (37) and , we haveBy the resultswe obtainFrom (37), (38), and (40), we have Choose , we getThus, similar to the proof of Theorem 7, the nosed-based algorithm (29) can solve the problem (29).
Remark 10. 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.
Remark 11. 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.
6. Simulations
Consider the following distributed optimization: where , Figure 1 is the communication topology. ChooseSimulation results employing edge-based algorithm (5) and node-based algorithm (29) are presented in Figures 2–4 and Figures 5–7, respectively. In Figures 2 and 5, it can be observed that the states of consensus are achieved. In Figures 3 and 6, the adaptive control gains first increase, then decrease, and finally converge to some constants, which is consistent with our theoretical analysis in Remark 8. In Figures 4 and 7, the evolution of is presented, which reaches the optimal value .
7. Conclusion
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.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
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).