Study On Distributed Consensus Optimization Algorithms Of Multi-agent Systems

With the improvement of important infrastructure such as cellular telephone networks,power grids and the Internet,and the development of corresponding techniques including communication technology and control theory,a large number of practical problems ultimately boil down to distributed consensus optimization in multi-agent systems.Distributed here is embodied in the modeling and algorithm implementation of optimization problems,which can only be accomplished through local nodes and cooperation with neighboring nodes.This non-central node characteristic of the multi-agent system network makes it difficult to obtain global optimization decision variables,which is the most fundamental difficulty to design distributed algorithms.This paper uses the celebrated Belief Propagation and powerful tools built on it to study distributed consensus optimization problems and propose novel distributed optimization algorithms.The main contributions and innovations are as follow:First,abandon the asymptotic consensus protocol,which is the conventional way of designing distributed optimization.In the case of acyclic and undirected connected graphs,based on the finitetime consensus from the Belief Propagation,design a distributed accurate Newton algorithm;Second,in the distributed primal-dual algorithm of acyclic and bi-directed connected graphs,the distributed information transfer strategy based on the Belief Propagation is used to decouple the linear equations through finite step iterations to obtain an accurate solution,and prove that the information transmission strategy used is equivalent to the Gaussian elimination method;Third,through appropriate modeling,the distributed consensus optimization problem is converted to meet the restricted form of the objective function of the celebrated Message-Passing algorithm,and then give a feasible distributed consensus optimization Message-Passing algorithm.Overall,the key technologies used in this paper to study distributed consensus optimization ultimately boil down to the finite-time consensus and the information transmission strategy which are both based on the Belief Propagation.The use of these tools allows us to design distributed optimization algorithms from a centralized perspective.The specific research content and research results include the following aspects in the order of chapters:1.Consider using gradient methods to study the distributed consensus optimization problems,we propose a distributed Newton optimization algorithm based on Belief Propagation.The global Newton gradient direction is constructed by finitetime consensus based on the Belief Propagation,and the overall design of the algorithm is completed by embedding it in the outer Newton iteration.The proposed distributed Newton algorithm,in the case of acyclic and undirected connected graphs,can be regarded as a distributed implementation of the centralized Newton method and has the consistent convergence;The proposed algorithm can also intercept a certain amount of inner iterations and extend it to general undirected cyclic connected graphs.Simulation shows that the algorithm still has good convergence.2.Consider using primal-dual methods to study the distributed consensus optimization problems,we propose a distributed primal-dual optimization algorithm based on Belief Propagation.Under the framework of primal-dual,we first transform a series of coupled sub-optimization problems on the primal iteration into a problem of solving linear equations,and then,within a limited step,use the distributed information transfer strategy based on Belief Propagation to accurately solve the equations,and finally build the distributed primal-dual Belief Propagation algorithm.As a comparison with the proposed algorithm,the distributed primal-dual generalized gradient algorithm is inductively proposed by the Jacobian iterative solution method of linear equations.The proposed algorithm based on the Belief Propagation,for acyclic undirected connected graphs,can be regarded as a centralized primal-dual algorithm;intercept a certain amount of inner iterations and extend the proposed modified algorithm to general undirected connected graphs,the simulation experiment shows that the proposed algorithm based on the Belief Propagation has linear convergence and has more advantages than the distributed primal-dual generalized gradient algorithm.3.Aiming at the distributed consensus optimization problem,through appropriate modeling,we propose a feasible Message-Passing algorithm.First,through introduce the well-known Message-Passing algorithm and its well-posedness and convergence,give an intuitive explanation of the Message-Passing algorithm;Secondly,through the modeling of the edge function,the distributed consensus optimization problem is equivalently transformed into a restricted form that satisfies the objective function of the Message-Passing algorithm;Finally,for the quadratic objective function,construct a feasible Message-Passing algorithm,in particular,the high-order nonlinear objective function only needs to go through the second-order approximation to use this algorithm.The simulation experiment verifies the feasibility and convergence of the proposed Message-Passing algorithm on the distributed consensus optimization problem.In short,this thesis focus on studying distributed consensus optimization problems.A series of new distributed optimization algorithms are proposed and extended to general undirected connected graphs.The proposed distributed optimization algorithm can be used as a supplement and improvement to the distributed convergence optimization results,and have important theoretical significance and wide application value.