矩阵方程的分布式求解算法研究概述

近年来,随着大规模网络的兴起和分布式优化理论的广泛应用,矩阵方程的分布式求解算法研究也受到了越来越多的重视.矩阵方程的计算求解在理论和工程领域都有着重要的意义.在多智能体网络下的分布式计算问题中,矩阵方程中的数据信息按照各种方式进行划分,单个智能体只能够获取其中的一份数据,然后通过与其邻居智能体进行信息交互,最终合作求解出不同类型的符合方程要求的解.本文集中讨论了近几年来针对线性代数方程、几类不带约束和带约束线性矩阵方程、以及其他矩阵相关的分布式计算和求解问题,介绍了投影一致方法、转化成分布式优化问题再求解的方法、以及针对特殊矩阵如稀疏矩阵的信息传递方法等分布式算法设计方法.最后,简要总结全文以及对分布式矩阵计算方向的研究进行了展望.

A survey of distributed algorithms for solving matrix equations

In recent years, with the rise of large-scale networks and the widespread application of distributed optimization theory, distributed algorithms for solving matrix equations have received increasing research attention. The computation of matrix equations is of great importance in both theoretical and engineering fields. In the distributed computation over multi-agent networks, the data information of matrix equations is partitioned in various ways. Each agent is able to obtain only one partition of the data and communicate with its neighbors, but all the agents can cooperatively solve different types of solutions as required. In this survey, we focus on the distributed algorithms in recent matrix computation problems, such as linear algebraic equations, several types of unconstrained and constrained linear matrix equations, and other matrix-related problems. We introduce distributed algorithms such as projection with consensus, distributed optimization transformation, and special methods such as message passing methods for sparse ones. Finally, we give a brief summary and an outlook on the research area of distributed matrix computation.