Research On Distributed Optimization Algorithms In Complex Communication Environments

In recent years,with applications and developments of wireless sensors,multi-robots and cognitive network,distributed optimization problems over networks have been attracted more and more attentions,and gradually permeated into scientific theory research areas,engineering applications and social life.In distributed optimization problems,the complex large-scale optimization problem of the whole network system will be assigned to multiple nodes for distributed optimization and computations.Here,”distributed” means that all the nodes do not require to know the global information,but only optimize and make decisions independently by using local information according to certain coordination mechanisms and rules in order to complete the system-wide optimization goal.There is an inseparable relationship between distributed optimization problems and network environments.How to use efficient network resources and deal with the change of network environments is an essential problem in the distributed optimization research area.This thesis focuses on distributed optimization problems under complex communication environments.The main contents are summarized as follows:(1).Distributed optimization problems based on the decentralized event-triggered communication are discussed.By designing decentralized state-dependent event-triggered conditions,an event-triggered Zero-Gradient-Sum algorithm(ZGS)is proposed.Based on the decentralized framework,the update of each node's state and the detection of each node's event-triggered conditions are no longer dependent on its neighbors' continuous information but only require the latest broadcast values at triggered instants,which further reduces the communications compared with continuous detections.The derivative of Lyapunov function is negative definite along the trajectories of the closed-loop system by appropriate choices of parameters in the event conditions.With the aid of La Salle's invariance principle,each node can be shown to converge to the optimal solution of the optimization problem and Zeno behavior is removed.For each node,a compact set is further defined and the subset of algorithms on this compact set are proved to be exponentially convergent.Moreover,a lower bound on convergence rate of these algorithms is given.(2).Distributed optimization problems based on periodic sampled-data are discussed.Firstly,a periodic ZGS algorithm based on sampled-data is presented.Then,by designing decentralized event-triggered conditions,a decentralized periodic event-triggered ZGS algorithm based on sampled-data is proposed.Based on the decentralized periodic event-triggered mechanism,the update of each node's state and the detection of each node's event-triggered conditions require its neighbors' latest broadcast values at triggered instants rather than its neighbors' periodic information at every sampling instants.Compared with periodic detections,less communications are required.Compared with continuous detections,less communications and less sampling are required.Since the event time interval is no less than the sampling period,the main advantage of this decentralized periodic event-triggered strategy lies in that Zeno behavior is naturally removed.The derivative of Lyapunov function is guaranteed to be negative definite along the trajectories of the closed-loop system by designing an appropriate relationship between the parameters involved in event-triggered conditions and the sampling period.Applying the La Salle's invariance principle,it is shown that each node can converge to the optimal solution of the optimization problem.Moreover,a subset of algorithms on some defined compact set are proved to be exponentially convergent and a lower bound on convergence rate of these algorithms is given.(3).The exponential convergence of ZGS algorithm over time-varying topologies are investigated.Firstly,a new connected condition on time-varying topologies,called cooperatively connected,is proposed.This condition does not require the topology constantly connected or jointly connected but only requires the integral of the Laplacian matrix of the network topology over a period of time is connected.Then,by establishing an important mathematics lemma,an approach which is based on the difference of Lyapunov function rather than its differential is put forward.Finally,it is proved that ZGS algorithms are exponentially convergent under the condition of cooperatively connected.(4).Distributed optimization problems based on event-triggered communications over timevarying topologies are studied and an event-triggered ZGS algorithm over time-varying topologies is proposed.Firstly,suppose the time-varying topologies are cooperatively connected.Then,the exponential decay time-dependent event-triggered conditions for each node are designed.The detection of this kind of event-triggered condition only depends on each node's own information and the external signal inputs rather than its neighbors' information at any time.Compared with the state-dependent continuous detections,the main advantage of this kind of event-triggered conditions is reducing communications among agents.By establishing of a key mathematics lemma and based on the method of the difference of Lyapunov function,the exponential convergence of the proposed algorithms is achieved.