Stochastic Primal-Dual Algorithms For Distributed Constrained Optimization

With the rise of artificial intelligence and big data,the technologies of sensing,communication and computing have developed rapidly.For large-scale optimization problems,centralized methods are time-consuming and may fail due to excessive burdens of computation and communication.In contrast,distributed optimization methods of multi-agent systems are more e cient,more reliable and can protect privacy,which have attracted much attention from researchers.Many problems in practice are usually constrained and accompanied by various random uncertainties such as the changes in the structure of communication networks,observation noises of function values and their gradients.Hence,it is of great theoretical significance and application prospects to study distributed algorithms for solving stochastic distributed equality and inequality constrained optimization problems.In this thesis,we mainly investigate two settings of stochastic constrained optimization problems.In the first setting,a stochastic convex optimization problem with both the objective function and the constraint functions are of expectation form is considered.Based on Lagrangian method and stochastic approximation,a primal-dual stochastic proximal method is proposed.Then,the almost sure convergence of the primal iteration sequence of the algorithm to the optimal solutions is proven under mild conditions.Besides,we analyze the upper bound of the expected function gap of the algorithm and the sublinear convergence rate is obtained with given step-size.Finally,a simulation example is given to justify to the convergence of the algorithm.In the second setting,a distributed stochastic optimization problem with equality constraints over a random network with imperfect communications is considered,where the measurements of all functions to be used are subject to noises.The goal of the network is to minimize a global cost function subject to a global constraint set,where the global objective is a sum of objective functions,the global constraint set is the intersection of local constraint sets with equality constraints,and each agent only has access to information on its cost function and constraint function.A primal-dual projection-free stochastic distributed algorithm is proposed to solve the problem.Almost sure stability,consensus and convergence of the algorithm are proven under mild conditions.Finally,a simulation example of distributed quadratic programming is provided to verify the theoretical results of the proposed algorithm.