Research On Prescribed-time Distributed Optimization Problem

In recent years,distributed optimization have a wide range of applications in various fields,such as formation control,smart grid,sensor network and distributed cooperative positioning.By utilizing algebraic graph theory,matrix theory,convex analysis theory and Lyapunov stability theory,the prescribed-time distributed optimization problems are investigated from the perspective of practical applicability.Firstly,under the agent’s state is unconstrained,the prescribed-time distributed optimization and Nash equilibrium solution problems are investigated from two different perspectives,respectively.Secondly,the prescribed-time distributed optimization problems with constraints are investigated.The results show that the algorithms can achieve convergence within any prescribed-time,when the initial states of the agents are given arbitrarily.Finally,simulation examples verify the effectiveness of the proposed algorithms.The research contents of this thesis mainly include the following aspects:1.Considering that the behavior between agents is cooperative,the problem of prescribedtime distributed optimization is investigated,where the optimization goal of each agent is to minimize the sum of all local objective functions within the prescribed-time.To address this problem,this thesis assumes that each agent can update their own states only by communicating with its neighbors.Then the prescribed-time distributed optimization algorithm is proposed based on the gradient descent algorithm and consensus algorithm.When the objective function is strongly convex,an appropriate Lyapunov function is selected to prove that the multi-agent system can converge to the optimal solution within the prescribed-time.The algorithm can achieve convergence within any prescribed-time,and the convergence time is fully independent of the initial conditions and system parameters.2.Considering that the behavior of the agent is based on its own interests,the problem of prescribed-time distributed Nash equilibrium seeking is investigated,where the optimization goal of each agent is to minimize its own objective function.To address this problem,this thesis assumes that agents can only update their own state by local information.Then the prescribed-time distributed optimization algorithm is proposed based on the gradient algorithm and leader-following consensus algorithm.When the objective function is strongly convex,an appropriate Lyapunov function is selected to prove that the multi-agent system can converge to the Nash equilibrium within the prescribed-time.The algorithm can achieve convergence within any prescribed-time,and the convergence time is fully independent of the initial conditions and system parameters.3.Considering that the state of each agent is constrained,the prescribed-time distributed optimization problem with constraints is investigated.In this problem,this thesis assumes that agents update their own states by communicating with its neighboring agents.Then the prescribed-time distributed optimization algorithm with constraints is proposed based on gradient projection algorithm and consensus algorithm.In addition,the constrained optimization problem is transformed into the unconstrained optimization problem,and the prescribedtime distributed optimization algorithm with constraints is proposed based on gradient descent algorithm and consensus algorithm.When the objective function is strongly convex,an appropriate Lyapunov function is selected to prove that the multi-agent system can converge to the optimal solution within the prescribed-time.The algorithms can achieve convergence within any prescribed-time,and the convergence time is fully independent of the initial conditions and system parameters.