Research On Solving Network Distributed Optimization Problem Using Continuous-time Algorithms

Decision-making,control and learning problems arising in many large-scale engineering systems can be modeled as distributed optimization problems in multi-agent networks,such as resource allocation in communication networks,target location in wireless sensor networks,economic dispatch problem in smart grids,distributed formation control in unmanned systems,machine learning in image recognition and so forth.In the context of multi-agent networks,the objective of distributed optimization is that the agents cooperatively find the optimal solution of the global objective function by using agents' local information and their neighbors' information.Recently,distributed optimization algorithms in multi-agent networks have received considerable attention.Especially,due to the demand of some practical systems in real time optimization and the advantages of control techniques in algorithm design and convergence analysis,continuous-time distributed algorithms have attracted more and more attention.Therefore,the study on continuous-time distributed algorithms has significant theoretical value and application prospect for network optimization.In continuous-time setting,this thesis studies a class of distributed convex optimization problems,where the total objective function is composed of a sum of some convex functions.The main research contents are in the following:(1)First,this thesis studies a distributed convex optimization problem subject to a general type of constraints,where each agent only knows its local objective function and local constraints including closed convex sets,equality and inequality constraints.To solve the problem in a distributed manner,the agents are allowed to communicate over a fixed and connected graph.First,a continuous-time distributed subgradient projection algorithm based on the Karuch-Kuhn-Tucker(KKT)condition of an equivalent optimization problem.By using the LaSalle invariance principle from nonsmooth analysis,it is proved that the agents' optimal variables reach consensus at an optimal solution.Second,differential projection dynamics and primal-dual projection dynamics are respectively designed based on saddle-point/primal-dual flow method,and the asymptotic stability of these two algorithms at their individual equilibrium points is shown.Finally,the performance of the proposed algorithms is illustrated by numerical simulations and their advantages and disadvantages are discussed.(2)Second,this thesis investigates an unconstrainted distributed convex optimization over a strongly connected directed graph.From the original problem and its equivalent Fenchel dual problem,continuous-time coordination algorithms with single time-scale and two time-scale are established,respectively.When the local cost functions are strongly convex with globally Lipschitz gradients,it is shown that the agents' optimal variables converge exponentially to the optimal solution.In particular,when the communication topology among agents switches over a set of strongly connected and weight-balanced directed graphs,the optimal solution can be also obtained with an exponential convergence rate under the proposed algorithms.Also,the theoretical results are further illustrated by numerical simulations.(3)Finally,this thesis considers a resource allocation problem for a group of agents communicating over a strongly connected directed graph,where the agents' local decision variables are subject to closed convex sets and satisfy some coupling equality constraints.With a strongly connected and weight-balanced directed graph,continuous-time distributed algorithms with projection feedback are first designed.The convergence analysis indicates that when the local objective functions are strongly convex,the agents' output states converge asymptotically to the optimal solution of the resource allocation problem.Particularly,the agents' decision variables converge exponentially to the optimal solution of the reduced resource allocation without closed convex sets.Moreover,continuous-time gradient algorithms are constructed over a strongly connected and weight-unbalanced directed graph for the reduced case without local convex sets.In this case,the gradient algorithms converge exponentially to the optimal solution of the reduced problem when the strong convexity and the Lipschitz continuity of the local cost functions and their gradients are individually satisfied.Moreover,numerical simulations verify and visualize the theoretical results.