Alternating Projection Method for Intersection of Convex Sets, Multi-Agent Consensus Algorithms, and Averaging Inequalities

IF 0.7 4区数学 Q3 MATHEMATICS, APPLIED Computational Mathematics and Mathematical Physics Pub Date : 2024-06-07 DOI:10.1134/s0965542524700155

A. V. Proskurnikov, I. S. Zabarianska

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Abstract

The history of the alternating projection method for finding a common point of several convex sets in Euclidean space goes back to the well-known Kaczmarz algorithm for solving systems of linear equations, which was devised in the 1930s and later found wide applications in image processing and computed tomography. An important role in the study of this method was played by I.I. Eremin’s, L.M. Bregman’s, and B.T. Polyak’s works, which appeared nearly simultaneously and contained general results concerning the convergence of alternating projections to a point in the intersection of sets, assuming that this intersection is nonempty. In this paper, we consider a modification of the convex set intersection problem that is related to the theory of multi-agent systems and is called the constrained consensus problem. Each convex set in this problem is associated with a certain agent and, generally speaking, is inaccessible to the other agents. A group of agents is interested in finding a common point of these sets, that is, a point satisfying all the constraints. Distributed analogues of the alternating projection method proposed for solving this problem lead to a rather complicated nonlinear system of equations, the convergence of which is usually proved using special Lyapunov functions. A brief survey of these methods is given, and their relation to the theorem ensuring consensus in a system of averaging inequalities recently proved by the second author is shown (this theorem develops convergence results for the usual method of iterative averaging as applied to the consensus problem).

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用于凸集交集的交替投影法、多代理共识算法和平均不等式

摘要在欧几里得空间中寻找几个凸集的公共点的交替投影法的历史可以追溯到著名的求解线性方程组的 Kaczmarz 算法，该算法设计于 20 世纪 30 年代，后来在图像处理和计算机断层扫描中得到广泛应用。I.I. Eremin、L.M. Bregman 和 B.T. Polyak 的著作对这一方法的研究起到了重要作用，这些著作几乎同时问世，其中包含了关于交替投影收敛到集合交点的一般结果，假定这个交点是非空的。在本文中，我们考虑的是凸集相交问题的一个修正，它与多代理系统理论有关，被称为约束共识问题。该问题中的每个凸集都与某个代理相关联，一般来说，其他代理无法访问。一组代理感兴趣的是找到这些集合的共同点，即满足所有约束条件的点。为解决这一问题而提出的交替投影法的分布式类似方法会导致一个相当复杂的非线性方程组，其收敛性通常使用特殊的 Lyapunov 函数来证明。本文简要介绍了这些方法，并说明了它们与第二位作者最近证明的确保在平均不等式系统中达成共识的定理的关系（该定理发展了应用于共识问题的通常迭代平均方法的收敛结果）。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Computational Mathematics and Mathematical Physics MATHEMATICS, APPLIED-PHYSICS, MATHEMATICAL

CiteScore

1.50

自引率

14.30%

发文量

125

审稿时长

4-8 weeks

期刊介绍： Computational Mathematics and Mathematical Physics is a monthly journal published in collaboration with the Russian Academy of Sciences. The journal includes reviews and original papers on computational mathematics, computational methods of mathematical physics, informatics, and other mathematical sciences. The journal welcomes reviews and original articles from all countries in the English or Russian language.