On Distributed Resource Allocation With Local Inequality Constraint And Local Set Constraint

The resource allocation problem is an essential optimization problem of the network system,and has been widely researched in many domains,such as sensor network,smart grid,transportation system and so on.Since the distributed algorithm is robust for the inaccuracy caused by the the failure of devices,external disturbances,time delay,does not require a central coordination unit to obtain the overall information of the optimization problem,and can protect the privacy of agents,it attracts much attention from the academic groups and industrial community.Based on the graph theory,nonsmooth optimization theory and Lasalle invariance principle,the relaxed resource allocation problem with communication delay,the resource allocation problem with local box constraint,the resource allocation problem with local inequality constraint,the resource allocation problem with local set constraint and the resource allocation problem with uncertain parameters are considered in this dissertation.For these problems,we design the corresponding distributed algorithms,and provide the convergent analysises of algorithms,respectively.1.The relaxed resource allocation problem with the heterogeneous communication delay is considered in this section.Under the assumption that the topology structure of the communication network is strongly connected directed graph and the communication delay is heterogeneous,we design a second-order distributed algorithm based on the optimality condition of the relaxed resource allocation problem.Moreover,based on graph theory and Lyapunov–Razumikhin stability theory,it is proved that the proposed algorithm can converge to the optimal solution.Finally,a numerical example of smart grid is given to demonstrate the effectiveness of the proposed algorithm.2.The resource allocation problem with the local box constraint is considered in this section.First,we adopt the exact penalty function method to transform the resource allocation problem with the local box constraint to an equivalent optimization problem without local constraint,employ the primal-dual theory to design a distributed resource allocation algorithm,which can converge to the optimal solution from any initial state,and then provide the convergence analysis of the proposed algorithm based on the nonsmooth optimization theory and Lasalle invariance principle.Second,we adopt the alternating direction multiplier method and finite-time consensus method to design a discrete-time distributed algorithm for the resource allocation problem.The proposed algorithm can apply to the resource allocation problem on not only the undirected graph but also the directed graph.Moreover,the proposed algorithm can ensure that the local constraints are satisfied during the whole computation process.3.The resource allocation problem with the local inequality constraint is considered in this section.The local cost functions of resource allocation problem is assumed to be convex and nonsmooth.To solve the resource allocation problem,a distributed continuous-time algorithm is designed based on the optimal condition of the resource allocation problem,and the convergence analysis is provided with the constructed Lyapunov functions and Lasalle invariance principle.Finally,we use an example of hybrid powerwater system to demonstrate the effectiveness of the proposed algorithm.4.The resource allocation problem with the local set constraint is considered in this section.We adopt the projection operator to design a distributed continuous-time algorithm for the resource allocation problem whose local feasible constraint is a convex set and global constraint is a coupled equation.Moreover,an extension problem of the resource allocation problem is also considered,whose global constraint is a coupled inequality.For two classes of problem,the local object function of each agent can not be globally convex,and is only required to be convex in its local constraint set.For the proposed algorithm,each agent only need exchange a little information on the ancillary variables with its neighbors,and thus the privacy of agents can be favorably protected.Moreover,the proposed algorithm has no requirement for the initial state of agents,which means that the algorithm can converge to the optimal solution from any initial state.5.The resource allocation problem with local uncertain set constraint is considered in this section.Based on the scenario theory,we first make each agent extract some samples of uncertain parameter from the uncertain set,and then transform the original resource allocation problem into a determinate optimization problem with a finite of set constraints.Next,we respectively provide the probability that the solution of determinate resource allocation problem is feasible for the original optimization problem,when all the agents possess a common scenario set and each agent has a private scenario set.Finally,we use a power system with wind power plants to demonstrate the research results.