通信噪音条件下非光滑优化问题的分布式derivative-free方法  

Derivative-free Distributed Methods with Communication Noises for Nonsmooth Optimization

作　　者：李国强 余淑辉 LI Guoqiang;YU Shuhui(School of Mathematics and Statistics,Guizhou University of Finance and Economics,Guiyang,Guizhou,550025,P.R.China;School of Big Data Statistics,Guizhou University of Finance and Economics,Guiyang,Guizhou,550025,P.R.China)

机构地区：[1]贵州财经大学数学与统计学院,贵州贵阳550025 [2]贵州财经大学大数据统计学院,贵州贵阳550025

出　　处：《数学进展》2024年第1期193-214,共22页Advances in Mathematics（China）

基　　金：Supported by Guizhou Provincial Department of Education’s Higher Education Scientific Research Project(No.[2022]172);Guizhou Province Science and Technology Plan Project(No.ZK[2022]General022);Universities Key Laboratory of System Modeling and Data Mining in Guizhou Province(No.2023013)。

摘　　要：本文研究了时变有向图上的非光滑分布式优化.在这样的图中,网络拓扑不仅是强连通的,而且还存在通信噪音.每个节点只能访问其非平滑的局部成本函数.本文给出了一种derivative-free分布式方法来最小化该网络中所有节点的成本函数之和.然后建立了所提出方法的收敛性分析,并获得了收敛速度的显式复杂性界限.当每个局部成本函数都是凸的时,我们的分析表明,所提出的算法以■的速率收敛,收敛速率取决于噪声的上限、光滑参数以及网络信息传播速度和节点间不平衡影响.当每个局部成本函数fi是强凸时,我们得到了O(lnt/t)的更快的收敛速度.最后,用一个数值实验来展示所提出方法的收敛性.In this paper,a nonsmooth distributed optimization over a time-varying directed graph is investigated.In such a graph,the network topology is not only strongly connected,but also there exist communication noises.Each node only has access to its local cost function which is nonsmooth.A derivative-free distributed method is proposed for minimizing the sum of cost functions of all nodes in this network.Then,the convergence of the proposed method is established and the explicit complexity bounds for the rate of convergence are also obtained.When each local cost function is convex,our analysis shows that the proposed algorithm converges at a rate of■,which depends on the upper bound of noises,the smoothing parameters,and both the speed of the network information diffusion and the imbalances of influence among the nodes.When each local cost function f_i is strongly convex,we obtain an improved convergence rate of O(lnt/t).A numerical example is used to show the efficiency of the proposed method.

关 键 词：分布式优化 凸优化 derivative-free算法 通信噪音 

分 类 号：O224[理学—运筹学与控制论]