基于原对偶乘子法的分布式优化
Distributed Optimization Using the Primal-Dual Method of Multipliers
摘要：
在本文中，我们提出了原始-对偶乘数法（PDMM）的分布优化图。特别是，我们优化的凸函数定义在一个图，其中图中的每一个边缘进行线性等式约束的总和。在设计新算法时，构造了一个增广的原始-对偶拉格朗日函数，它可以平滑地捕捉图的拓扑结构。结果表明，鞍点的构造函数提供了原问题的最优解。在同步更新和异步更新两种情况下，PDMM对一般闭函数、真函数和凸函数都具有O（1/K）的收敛速度（K表示迭代次数）.通过分布平均实验，研究了PDMM的其他性能，如不同参数设置下的收敛速度和对传输失败的恢复能力。
英文摘要：
In this paper, we propose the primal-dual method of multipliers (PDMM) for distributed optimization over a graph. In particular, we optimize a sum of convex functions defined over a graph, where every edge in the graph carries a linear equality constraint. In designing the new algorithm, an augmented primal-dual Lagrangian function is constructed which smoothly captures the graph topology. It is shown that a saddle point of the constructed function provides an optimal solution of the original problem. Further under both the synchronous and asynchronous updating schemes, PDMM has the convergence rate of O(1/K) (where K denotes the iteration index) for general closed, proper and convex functions. Other properties of PDMM such as convergence speeds versus different parameter- settings and resilience to transmission failure are also investigated through the experiments of distributed averaging.
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