Distributed Push-sum Subgradient Optimization Algorithm Of Multi-agent Systems

In recent years,multi-agent system has widely applications in artificial intelligence, creature, network communication, satellite positioning.Thus, it has gained more and more attentions from scholars of different fields. With the development of science,more sensitive reaction, more simply operated method of distributed processing received widespread attention.As a result,,the distributed optimization algorithms of multi-agent network has been developed rapidly. The distributed convex optimization algorithm can solve large-scale convex optimization problems based on the communication among agents of multi-agent networks.But if each agent in the network interacts information at the same time, it is easy to cause the information congestion, which can make the information not arrive at the receiving end timely, and thus the whole network may be lack of robustness. It is therefore more important to solve the above problems when investigates the optimization problems of multi-agent networks.This paper mainly studies the optimization problems in directed networks, where the information communication between agents maybe prone to congestion. The main work of this paper includes the following two parts:Firstly, this paper studies the distributed Push-sum subgradient optimization algorithm in the asynchronous communication of the networks.For the synchronous communication in the directed networks,all the agents communication with their neighbors at the same time,which makes the path with lots of information,then leading to congestion even collapse. Thus,we think of the two agents of the networks communication with each other at a certain time,and the others of the networks keep the information of the last moment,until the agents of system achieve consistently. This method avoid the information congestion at radically,but it will delay the convergence rate of the agents.lt is shown that the asynchronous Push-sum subgradient optimization algorithm convergence,under the assumptions that the subgradient of each agent’s objective function is bounded and the stochastic time-varying directed network is uniformly strongly connected,and its rate is O(tne-kt+Int/(?)t).The rate is depend on the number of the system and the probability of the selected agents,what is more,the rate is slower than the synchronous communication.Secondly,we study the distributed Push-sum subgradient optimization algorithm with time-varying communication delay.For the synchronous communication of the directed networks result in congestion or the reality path is damaged or other reasons,we put forward the Push-sum subgradient optimization algorithm with time-varying communication delay.It is difficult to analysis the algorithm with time-varying communication delay,so this paper studies its convergence by system augment which convert the delay case to the non delay case.In the existed system augment,the diagonal elements of the adjacency matrix are all positive.But the system augment of this paper does not require that,maybe they are zero.Under the assumptions that communication delays and the subgradients are bounded, and the switching directed networks are periodically strongly connected, we prove that the convergence of the proposed distributed Push-sum subgradient optimization algorithm.But it is shown that the convergence rate in the case of communication delays is slower than that without communication delays.Finally,the conclusion is verified by numerical simulation.In summary, under the assumptions that the subgradient of each agent’s objective function is bounded and the stochastic time-varying directed network is uniformly strongly connected, this paper proves the convergence of the asynchronous Push-sum subgradient optimization algorithm, Moreover, under the assumptions that communication delays and the subgradients are bounded and the switching directed networks are periodically strongly connected, the convergence of the proposed distributed Push-sum subgradient optimization algorithm is also presented.lt is shown that the asynchronous communication and the communication delays can still ensure the convergence of the optimization algorithm, but with a slow convergence rate.