A Study On Consensus-based Distributed Robust Convex Optimization Problem

This dissertation studies the consensus-based distributed learning of robust convex optimization problems.For robust convex optimization problems,it is difficult to solve numerically due to the uncertainty in its constraints.However,SA(Scenario Approach)was proposed to provide a solution to this problem.SA transfers the original problem to its probability approximation problem SP(Scenario Problem)by sampling a finite number of constraints from the constraints of the original problem according to i.i.d,and the obtained SP is a standard solvable convex optimization problem.Obviously,the greater the number of SA samples,the higher the probability of SP approximating the robust convex optimization is,which will also lead to the the increase in the number of samples and in the demanding performance requirements for a single processor.Nevertheless,distributed learning can solve such problem with large-scale samples and will distribute the large amount of samples to multiple nodes for processing and analysis.It is regulated that there is no interaction between the samples assigned to each node.In the process of analysis and processing,variables of nodes are exchanged through the network composed of nodes to achieve consensus.Based on the above background,this paper discusses the consensusbased distributed robust convex optimization within three aspects: DACbased(Distributed Average Consensus)distributed learning for robust convex optimization,Master-based ADMM(Alternating Direction Method of Multipliers)distributed learning for robust convex optimization and Slavebased ADMM distributed learning for robust convex optimization.1.DAC-based distributed learning for robust convex optimization.Firstly,we transfer the robust convex optimization to SP and model the distributed SP.Then,each node solves the subproblem of distributed SP with the distributed samples.Finally,the DAC strategy ia applied to consensus the solution obtained by each node.Considering that nodes in the network updating its variables only need the variables of its neighbors,while they are required to wait for all nodes to complete their previous updates to update their next updates in DAC,which will increase the overall computing time.Therefore,the EDAC(Efficient DAC)distributed learning algorithm based on the colored network is proposed.The characteristics of the colored network make the nodes with same color update their variables parallelly.This method can not only guarantee that its final result is consistent with the result of DAC,but also can greatly reduce the computing cost and shorten the computing time.2.Master-based ADMM distributed learning for robust convex optimization.Considering that the consensus results achieved by the DACbased strategy are flawed in accuracy,this is because that the consensus results may not be the optimal solutions despite that the solutions of all nodes optimise the subproblems,which causes the low accuracy of the final consensus results.Therefore,ADMM,which has higher accuracy guarantee,is proposed to distributedly learn the distributed SP.The ADMM distributed learning based on the master node is working on the network composed of Master-Slave nodes.The Master node controls the consensus of variables of all nodes in the network,and Slave nodes do not have variables interaction and only communicate with the Master node.According to the probabilistic approximation of the robust convex optimization,we construct a distributed SP model based on master node,which is solved by using a consensus-based distributed ADMM.3.Slave-based ADMM distributed learning for robust convex optimization.In the Master-based distributed learning,we find that the Master node is useless in the variable updates.Thus,we propose to adopt Slavebased distributed learning and obtain the consensus results by connecting nodes with its neighbors and allowing the variables interaction between nodes and its neighbors.After transferring the robust convex optimization to SP,we propose a distributed SP model based on slave nodes and applies ADMM to perform distributed solution.the convergence validity of the method are proved theoretically.Finally,it is also compared with the previous distributed learning method in the experiment.The results show that the proposed method is better than the DAC-based distributed learning method.