非平衡有向网络下带约束的连续时间分布式优化算法设计

Continuous time with constraints in general directed networks distributed optimization algorithm design

本文基于权重不平衡有向网络,对一类分布式约束优化问题进行研究,其中全局目标函数等于具有李普希兹梯度的强凸目标函数之和,并且每个智能体的状态都有一个局部约束集.每个智能体仅知道自身的局部目标函数和非空约束集.本文的目标是用分布式方法求解该问题的最优解.针对优化问题,提出了一种新的分布式投影梯度连续时间协调算法,利用拉普拉斯矩阵的零特征值对应的左特征向量消除了图的不平衡性.在某些假设下,结合凸分析理论和李雅普诺夫稳定性理论,证明了算法能够获得问题的最优解.最后,通过仿真验证了算法的有效性.

This paper studies a class of distributed constrained optimization problems on weighted unbalanced directed networks,in which the global objective function is equal to the sum of strongly convex objective functions with the global Lipschitz gradient,and the state of each node is limited to a local constraint set.Each agent only knows its own local objective function and the non-empty constraint set.The goal of this paper is to solve the optimal solution of the problem by using a distributed method.For the optimization problem,a new distributed projection gradient continuous-time coordination algorithm is proposed,in which the imbalance of the graph is eliminated by using the left eigenvector corresponding to the zero eigenvalue of the Laplace matrix.Under some assumptions,combined with the convex analysis theory and Lyapunov stability theory,it is proved that the algorithm can obtain the optimal solution of the problem.Finally,the effectiveness of the algorithm is verified by simulations.