权重不平衡有向网络下的分布式优化算法  

Distributed optimization algorithms over weighted unbalanced directed networks

作　　者：夏俊 倪伟 XIA Jun;NI Wei(Department of Mathematics,Nanchang University,Nanchang 330031,China)

机构地区：[1]南昌大学数学系,江西南昌330031

出　　处：《南昌大学学报（理科版）》2022年第3期303-308,313,共7页Journal of Nanchang University(Natural Science)

基　　金：国家自然科学基金资助项目(61663026);江西省自然科学基金资助项目(20192BAB207025)。

摘　　要：研究了有向网络下的分布式优化问题,其中每个智能体的局部目标函数的和构成了网络的全局目标函数。本文利用梯度跟踪和比例积分的策略对梯度的平均值进行跟踪,并设计变量对拉普拉斯矩阵零特征值的左特征向量进行跟踪,从而在权重不平衡有向网络下提出了一类基于梯度的固定步长分布式优化算法。将分布式优化算法从无向图推广到了有向图。在局部目标函数和其梯度分别满足强凸和李普希兹连续的情况下,结合凸分析和李雅普诺夫稳定性理论分析算法的收敛性,结果证明所提出的算法能够收敛到优化问题的最优解。The problem of distributed convex optimization under directed networks was investigated in this paper,where the global objective function of networks consists of the sum of the local objective functions of each agent.The gradient tracking and proportional integration strategies were used to track the average value of the gradients and auxiliary variables were introduced to track the left eigenvector of the zero eigenvalue of the Laplacian matrix.A class of gradient-based distributed optimization algorithms with fixed step size was proposed under weighted unbalanced directed networks.The distributed optimization algorithm was extended from undirected graph to directed one.When the local objective functions and its gradients satisfy strong convexity and Lipschitz continuity respectively,the convexity analysis and Lyapunov stability theory were combined to analyze the convergence of the proposed algorithm.The results showed that the proposed algorithm can converge to the optimal solution of the optimization problem.

关 键 词：分布式优化 有向图 多智能体系统 

分 类 号：TP316.4[自动化与计算机技术—计算机软件与理论] O231.4[自动化与计算机技术—计算机科学与技术]