Distributed Optimization For Multi-agent Systems Under Constraints

In recent years,the control problem of networked systems has become a hot research field.Compared with traditional centralized algorithms,the data of center-free distributed algorithms disperses across the entire network.It is widely used in smart grid,machine learning,cloud computing,sensor networks,UAVs and so on due to its distributiveness,extendibility,autonomy and robustness.In the actual situations,agents often subject to a variety of conditions.How to find the optimal decision of the network within the constraint sets has important practical significance.Therefore,the distributed optimization of multi-agent systems under all kinds of constraints has great practical value.The main contributions of this paper are summarized as follows:We design an initialization-free continuous time distributed algorithm to solve the optimization problem with local inequality constraints and constraint sets.Each agent has a local cost function,and needs to find the same global optimal solution within the intersection of the feasible sets.Firstly,using the penalty function method and the properties of the Sigmoid function,the inequality constraint is transformed into an equality constraint,which eliminates the limitation on the initial conditions of the agent.Based on the saddle point dynamics,we design a distributed continuous time algorithm.Through the Lyapunov stability analysis and the LaSalle's invariance principle,it is proved that the algorithm converges to the same optimal solution.Finally,the designed algorithm is applied to the problem of ridge regression analysis,which achieves goof effectiveness.We propose a first-order extremum-seeking algorithm to solve the resource allocation problem,where the specific expression form and gradient information of the local cost functions are unknown.Agents take advantage of measurements of local cost functions to find optimal allocation plan,where agents exchange estimated decisions with their neighbors under an undirected and connected graph.Making use of the Lyapunov stability theory and the average analysis method,the convergence of the proposed algorithm to the neighborhood of the optimal solution is presented.Then,the first-order algorithm is extended to the second-order algorithm with low-pass filters,which achieves better convergence performance than the first-order algorithm.Finally,the effectiveness of the proposed algorithms is illustrated by numerical examples and the proposed algorithms are applied to economic dispatch in smart grids,which achieves goof effectiveness.