Research On Distributed(Online) Constrained Optimization Algorithms Based On Multi-agent Networks

With the scale and complexity of modern network control system and the rapid development of modern communication technology,the traditional centralized optimization relies on a single control center to collect the information of the whole network system,so it is difficult to adapt to the needs of efficient,flexible,low-cost and safe operation of large-scale complex systems.In view of this,the distributed optimization based on multi-agent network came into being.In recent years,distributed optimization has been widely used in many fields,such as wireless sensor network,power grid system,resource allocation network,multi-robot system,machine learning and so on,so it has attracted more and more scholars' research and attention.Based on algebraic graph theory,convex optimization theory and multi-agent cooperative control theory,this paper studies the distributed(online)constrained optimization problem in the framework of networked multi-agent system.As the optimization decision variables of practical system optimization are usually constrained by various external and internal factors,compared with the unconstrained optimization problem,the constrained optimization problem is more general and complex,and has more practical research significance.This thesis focuses on the distributed constrained optimization and distributed online constrained optimization of multi-agent networks.The main work and contributions of this thesis are as follows:1.The problem of smooth distributed constrained optimization with multi-agent coupling equality constraints is studied.For different communication topologies,three distributed constrained optimization algorithms based on alternating direction multiplier(ADMM)are proposed.Firstly,under the condition of fixed(time-varying)undirected communication topology,the optimization problem with non-quadratic local objective function and local inequality constraints is solved by using center-free algorithm and ADMM algorithm.When a fixed(time-varying)undirected graph is a connected(joint connected)graph,the convergence of the algorithm and the optimality of the final convergence value are strictly proved.Then,under the fixed unbalanced digraph,using ADMM algorithm,Newton Raphson method and proportional consistency algorithm,the smooth distributed constrained optimization problem under the coupling equality constraint of multi-agent is studied.A robust optimization algorithm is proposed by considering the directed communication topology with communication delay and packet loss.Under the condition that the digraph is strongly connected and the corresponding adjacency matrix is column-stochastic,the convergence and the optimality of the convergence results of the two algorithms are strictly proved.2.The problem of nonsmooth distributed constrained optimization with multi-agent coupling equality constraints is studied.For different initial conditions,convexity conditions and network connectivity conditions,three continuous-time distributed constrained optimization algorithms are proposed.Firstly,under the condition of fixed undirected graph,the optimization problem with nonsmooth,general convex rather than strictly convex local objective function is solved by using nonsmooth analysis,differential inclusion theory and algebraic graph theory.The convergence results of the algorithm depend on the specific initial conditions.Therefore,a fully distributed initialization-free algorithm is proposed.By introducing auxiliary variables,the dependence of algorithm results on specific initial conditions is eliminated.At the same time,the algorithm does not need to execute any additional initialization procedure,which saves the cost of calculation and communication,and improves the flexibility of algorithm application.Finally,for the optimization problem of the balanced digraph and the local objective function being nonsmooth and strongly convex,the distributed algorithm is given and the convergence of the algorithm is proved.3.The problem of distributed constrained optimization with multi-agent coupling inequality constraints is studied.According to different communication topology conditions,two distributed constrained optimization algorithms are proposed.Firstly,a discrete-time optimization algorithm is proposed based on the projection primal-dual subgradient method and consistency strategy for the optimization problem with coupling inequality constraints and local set constraints over a fixed unbalanced digraph.When the communication graph is strongly-connected with row-stochastic matrix and the step size satisfies the given conditions,the optimal solution can be obtained asymptotically by using the algorithm.Secondly,a continuous-time optimization algorithm based on push-sum strategy and primal-dual subgradient method is proposed to solve the optimization problem in time-varying unbalanced digraph.When the communication graph is a joint strongly connected,it is proved that this kind of algorithm can obtain the optimal solution asymptotically.4.The problem of distributed online constrained optimization with multi-agent set constraints is studied.Two distributed online constrained optimization algorithms are designed according to whether the gradient information of the objective function is known.When the gradient information of the local objective function is unknown,a stochastic difference estimator is constructed based on the idea of Kiefer Wolfowitz algorithm.Meanwhile,dynamic regret is used to analyze and measure the convergence performance of the two online optimization algorithms.The theoretical analysis shows that when the growth rate of the deviation of the reference sequence is within a certain range,the upper bounds of dynamic regrets for the two online algorithms increase sublinearly with respect to the learning time.