Continuous-time Algorithms For Two Kinds Of Distributed Constrained Optimization Problems

With the development of the information age,distributed optimization problems have received increasing attention,especially in the areas of cloud computing and big data.In recent years,distributed optimization problems appear over many fields such as engineering application,scientific research and society.For distributed optimization problems,consensus optimization algorithms have become an important means of researching such problems and play a significant role in real applications.But it is a challenging work to design consensus optimization algorithms with lower complexity structures for distributed optimization problems.The objective function of the distributed optimization problems considered in this paper can be described as the sum of local convex objective functions.Based on the properties of multi-agent networks,two kinds of constrained distributed optimization problems are studied and related continuous-time algorithms with lower complexity structures are proposed.The main results are as follows:In the second chapter,a minimax distributed convex optimization problem is discussed.First of all,based on the approximation theory of minimax function,the minimax distributed convex optimization problem is approximated by the approximation distributed optimization problem.Next,a continuous-time optimization projection algorithm based on multi-agent network is proposed for the approximation distributed optimization problem.The algorithm can be described as a dynamic system.Moreover,the optimal solution problem of the approximation distributed optimization problem is converted into the convergence problem of the dynamic system.Finally,by Lyapunov method,we proved that the dynamic system is stable and the state solution is capable of reaching the optimal solution of approximation distributed optimization problem.And to illustrate the effectiveness of the algorithm,two numerical simulations are proposed.In the third chapter,a distributed convex optimization problem with equality and inequality constraints is studied.Based on connected and undirected multi-agent system,a continuous-time optimization algorithm based on Karush-Kuhn-Tucker condition is proposed.In the multi-agent network,the agents connect each others as an undirected graph and know only their own objectives and constraints.Compared with the existing continuous-time algorithm,algorithm provided in this chapter has the advantages of lower complexity structures and easy to implement.The algorithm can be described as a dynamic system.Moreover,the stability and convergence of the dynamic system are analyzed by Lyapunov method and we proved that the state solution of the dynamic system converges to the optimal solution to the optimization problem.Finally,a numerical example is given to illustrate the practicality of the algorithm.