Distributed Random Optimization Of Multi-agent Systems Over A Time-varying Directed Topology

With the rapid development of the networked multi-agent system,how to formulate optimization strategy in large-scale,open and dynamic information interactive environment has become a hot research topic.Information interaction between the agent based on the network topology,energy consumption and changes in the communication environment will lead to the change of the communication link,therefore,the existed research on fixed connected network topology is theoretical.Optimization under a time-varying to topology has more practical meaning.With the development of economy,the improvement of management level and the expansion of management vision.It becomes a social demand to solve the management optimization problems of large scale and large scale.Such as large-scale network traffic system,network communication system,power grid dispatch,economic game,logistics system,artificial intelligence and so on.It is of great research significance and practical application value to consider the optimization and decision-making of large complex multi-agent system under network topology.The consensus mechanism provides a method for decision-making of interactive behaviors among large distributed intelligence.Considering the change of energy consumption and communication environment of intelligent node,the connection of communication links are changed.Therefore,it is more practical significance to consider the optimization of time-varying topology.In this paper,the distributed constrained optimization problem is considered.The main research work includes:We propose a distributed stochastic gradient-free optimization algorithm for constrained optimization under time-varying directed topology.For cost function,we consider a general of nonsmooth cost function.Gaussian smoothing method is used to approximate the nonsmooth cost function,the expectations of the gradient-free of the original nonsmooth functions is equal to approximated the gradient of smooth function.The optimal solution of the approximate smooth function optimization problem is equivalent to the original nonsmooth function optimization problem.It overcomes the problem that the gradient of non-smooth function cannot be solved,and the calculation of subgradient is very difficult.At the same time,in view of the inevitable noise phenomenon in the communication link,the distributed stochastic gradient-free optimization algorithm is proposed considering the noise interference in the system modeling.By means of the upper martingale convergence theorem,it is proved that the state decisions of all agent are almost surely achieve consistency.The method of solution exists guarantee that decisions almost surely converge to the stationary optimal point.In an incomplete information game,players can only know part of the information of other players.In order to solve the general noncooperative Nash game problem of incomplete information in time-varying directed topology,we propose a broadcast gossip stochastic optimization control protocol.The network topology under broadcast gossip mechanism is a special kind of time-varying topology,which a player(nodes)randomly wakes,and waked player broadcast information to it's neighbor players.The communication weight matrix of broadcast network topology is the row random matrix.In this paper,the corresponding broadcast random consensus protocol is designed,and the upper martingale convergence theorem proves that all players can effectively estimate the decisions of all other players.Different with the aggregate games,the generalized noncooperative Nash games is studied in our paper,where the cost function for each player is related to the decisions of all other players.Based on convex analysis and variational inequality,the Nash equilibrium of a non-cooperative game is equivalent to the optimal solution to the constraint problem.Every player updates the decision of itself to minimize a cost function based on gradient method and the decision estimates of other players.It is shown that every player estimates the decisions of other players effectively and decisions almost surely converge to the Nash equilibrium by the broadcast gossip algorithm.