Design Of Distributed Optimization Protocol For Multi-agent Systems Over Directed Graphs

With the continuous development and application of the distributed artificial intelligence and large-scale complex intelligent systems,the optimization problem of the multi-agent system is becoming more and more complex and its scale is becoming larger and larger.The advantage of solving this problem by distributed method is becoming more and more obvious.Compared with the centralized control optimization strategy,the distributed method has the high performance of reducing communication load,protecting data privacy and improving system robustness.At present,distributed optimization algorithms are widely used in distributed positioning and tracking,large-scale machine learning,intelligent internet of things,coordinated control of robots,traffic control,virtual reality and so on.Therefore,the design and analysis of multi-agent distributed protocols become a popular research in the field of artificial intelligence.This paper studies the design of multi-agent distributed optimization protocols based on the directed topology in different situations.The main research contents are summarized as follows:For the distributed optimization problem of discrete-time systems under weighted balanced digraphs,a similar discrete-time sub-gradient distributed optimal control algorithm is proposed by referring to the idea of the closed-loop control input for continuous-time systems.Firstly,the optimization problem can be equivalent to the stability problem of the system by state transition.By constructing a Lyapunov function,the convergence and convergence rate of the algorithm are analyzed,and the sufficient condition for an exponential convergence are obtained.In addition,in order to reduce the number and burden of the communication,a distributed optimization protocol is improved with the idea of event triggering,and event triggering conditions for the agent state transfer are obtained.For the distributed optimization problem of the multi-agent system under a weighted unbalanced digraph,the fixed step size and row stochastic adjacency matrix are used to design the distributed convex optimization algorithm.In order to reduce the burden of information storage and avoid increasing the error caused by the gradient of the previous moment,the gradient difference strategy is not used in the protocol.The fixed step size accelerates the convergence of the optimization iteration,and the convergence rate of the algorithm is approximately linear.For the designed distributed optimization algorithm,the existence of the optimal solution is analyzed.The stability and convergence properties of the discrete-time algorithm are proved by means of Lyapunov function and mean value theorem.The nonsmooth convex optimization problem over weighted unbalanced digraphs is studied and a distributed randomized gradient-free optimization protocol designed with diminishing step sizes is given.In the algorithm,the network topology used in communication adopts the row stochastic matrix which is easier to acquire and implement in practice.The step size is chosen as a fixed form of diminishing step sizes,and the agent's cost function is a non-smooth function.The consistency,convergence and convergence rate of the algorithm are systematically analyzed and demonstrated by the idea of boundedness and limit.In the demonstration,the martingale convergence theorem is avoided,so that the requirement of diminishing step sizes is relaxed.The step size only needs to satisfy the condition of non-summability.Finally,the convergence rates under different step sizes are given.