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arXiv 1809.06008math.OC
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具有不同耦合源的多智能体optimization的酉分布次梯度方法

A unitary distributed subgradient method for multi-agent optimization with different coupling sources

Changxin Liu, Huiping Li, Yang Shi

备注：
15 pages, 2 figures
发表信息：
Automatica, vol. 114, pp. 1-12, 2020
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摘要：

在这项工作中，我们首先考虑分布式凸约束最优化问题的目标函数是由多个本地和可能的非光滑目标私人持有的一组代理编码，并提出了一个分布式次梯度方法与双平均（简称${\rm DSA_2}$），只需要点对点通信和本地计算来解决全局问题。算法框架建立在对偶方法和动态平均共识的基础上;通过迭代最小化总体目标的局部对偶模型来形成测试点序列，其中系数，即，目标的近似次梯度由动态平均一致性方案提供。我们从理论上证明了${\rm DSA_2}$具有非遍历收敛性，即，局部最小化序列本身是收敛的，这是在现有结果中找不到的明显特征。具体来说，我们建立了一个收敛速度的$O(\frac{1}{\sqrt{t}})$的目标函数误差。然后，将${\rm DSA_2}$与对偶分解相结合，将其推广到具有耦合函数约束的分布式最优化问题。这是可能的拉格朗日松弛变换的原始问题的约束耦合到对偶的费用函数，从而使我们能够通过${\rm DSA_2}$解决对偶问题。证明了对偶目标误差和二次罚函数均以$O(\frac{1}{\sqrt{t}})$的速度收敛，原目标误差渐近消失。数值实验和比较表明，所提出的算法的优势，并验证我们的理论研究结果。

英文摘要：

In this work, we first consider distributed convex constrained optimization problems where the objective function is encoded by multiple local and possibly nonsmooth objectives privately held by a group of agents, and propose a distributed subgradient method with double averaging (abbreviated as ${\rm DSA_2}$) that only requires peer-to-peer communication and local computation to solve the global problem. The algorithmic framework builds on dual methods and dynamic average consensus; the sequence of test points is formed by iteratively minimizing a local dual model of the overall objective where the coefficients, i.e., approximated subgradients of the objective, are supplied by the dynamic average consensus scheme. We theoretically show that ${\rm DSA_2}$ enjoys non-ergodic convergence properties, i.e., the local minimizing sequence itself is convergent, a distinct feature that cannot be found in existing results. Specifically, we establish a convergence rate of $O(\frac{1}{\sqrt{t}})$ in terms of objective function error. Then, extensions are made to tackle distributed optimization problems with coupled functional constraints by combining ${\rm DSA_2}$ and dual decomposition. This is made possible by Lagrangian relaxation that transforms the coupling in constraints of the primal problem into that in cost functions of the dual, thus allowing us to solve the dual problem via ${\rm DSA_2}$. Both the dual objective error and the quadratic penalty for the coupled constraint are proved to converge at a rate of $O(\frac{1}{\sqrt{t}})$, and the primal objective error asymptotically vanishes. Numerical experiments and comparisons are conducted to illustrate the advantage of the proposed algorithms and validate our theoretical findings.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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