Research On The Distributed Optimization Algorithm For The Convex Optimization With Globally Coupled Constraints

Many optimization problems in bio-medicine,physics,sociology,etc.,are large-scale and involve the processing of big data and multiple constraints.Traditional centralized algorithms cannot be used for solving this problem,which leads to the emergence of distributed optimization algorithms.The existing works on the distributed optimization mainly focus on the unconstrained convex optimization and the convex optimization without globally coupled constraints,which leads to an insufficiency in research on the convex optimization problems with globally coupled constraints.Therefore,it is of great significance in theory and applications to investigate the distributed convex optimization problems with globally coupled constraints.To this end,this dissertation focuses on the convex optimization problem with globally coupled constraints.The main contexts are listed as follows:For the optimization problem with global multiple constraints,we convert it into problem with with globally coupled constraints based on the characteristic of Laplacian matrix.Based on the projected primal-dual method and compensation variables,a fully distributed algorithm is proposed,which can obtain the optimal solution of the optimization problem under the condition that each of the multiple constraints is privatively known by each agent.With the help of the convergent sequences,the algorithm is proved to converge to the optimal solution of the optimization problem under certain sufficient condition with respect to the step-size.Numerical example is given to verify the efficacy of the proposed algorithm.Comparisons with the centralized method show that the computation complexity is largely reduced.For the optimization problem with coupled objective functions and coupled constraints,we design a distributed discrete-time algorithm.By employing compensation variables,the global optimization problem can be solved without the information exchange of coupled constraints.The convergence analysis of the proposed algorithm is presented with the convergence condition through which a diminishing step-size with an upper bound can be determined.Moreover the convergence rate analysis is given.The effectiveness and the practicability of the proposed algorithm is demonstrated by the parameter optimization problem in smart building.For optimization problem with coupled equality constraints,a novel distributed predefined-time convergent algorithm is proposed.Based on a distributed parameter learning method,by employing nonhomogeneous functions with exponential terms,the proposed algorithm can achieve a predefined-time convergence rate under the condition that the coupled constraints are satisfied during initialization.Application to the economic dispatch problem verifies the result,which demonstrates that the convergence rate of the proposed algorithm far outweighs that of current fixed-time convergent algorithms.For optimization with coupled equality constraints,an accelerated saddle point dynamics is firstly proposed over a second-order multi-agent network.Specifically,an inertial fast-slow dynamical system with vanishing damping is introduced to model this distributed saddle point algorithm where the dual variables are updated in two time scales,i.e.the fast manifold and the slow manifold,which leads to the fast convergence of the proposed algorithm.Application to the economic dispatch problem verifies the result,which demonstrates that the convergence rate of the proposed algorithm far outweighs that of the conventional algorithms.Comparisons show that the convergence rate of the proposed algorithm outweighs that of current the existing asymptotically convergent and exponentially convergent algorithms.For optimization with coupled equality constraints,by introducing the indirect dual variables,an indirect dual ascent method with an exponential convergence rate is proposed,in which the dual dynamics can be executed in a decentralized manner by all nodes over the network.Moreover,the exponential convergence rate of the proposed algorithm is established through the Lyapunov method and the Singular Perturbation Theory.In contrast to the conventional methods,consensus over all the dual variables are not required,this further leads to the simple dual dynamics,reduced communication burden and better privacy preserving.Application of the dynamic economic dispatch problem verifies the effectiveness and performance of the proposed algorithm.Comparisons show that the convergence rate of the proposed algorithm outweighs that of current the existing asymptotically convergent and exponentially convergent algorithms and its information exchange is reduced.