非线性算子零点的分裂算法：收敛性分析与应用

Splitting algorithms for treating zero points of nonlinear operator, which is a popular research topic of common interest in two areas of nonlinear analysis and operation research. The research topic, which finds a lot of applications in mechanics, economics, transportation and medical science, has a close relationship with nonlinear programming, matrix theory, convex analysis, and variational analysis and so on. The research topic is a common hot issue between nonlinear analysis and optimizatioin theory. This project aims to study, in the framework of reflexive Banach spaces, the convergence analysis of splitting algorithms and their applications, which in detail are as follows: (1) with the aid of the Bregman projection and the technique of mean iterations, design the algorithms for zero points of the sum of two monotone operators so that they have some properties of global convergence, stability and efficiency; (2) apply the algorithms to solve the Ky Fan inequality and minimizer problems of lower semicontinuous convex functions. This project has scientific significance and practical values for not only providing new theory and methods for solutions of nonlinear operator equations, but also offering new elements for the cross areas of several subjects.

非线性算子零点的分裂算法研究是非线性泛函分析与运筹学的一个交叉研究领域，它与非线性规划、矩阵理论、凸分析和变分分析等分支有着紧密的联系，在力学、经济、交通、医学等领域有着广泛的应用，是现代非线性分析与优化理论研究的重要课题之一。本项目旨在分裂算法在自反Banach空间框架下的收敛性分析和应用性研究，其研究内容主要包括：(1) 借助Bregman投影和均值技巧，设计两单调算子和算子的零点逼近算法，使之具有全局收敛性、稳定性、快速性；(2) 运用效的数值算法求解一Ky Fan不等式的解及下半连续凸函数最小值问题。本项目的实施能够为非线性算子方程求解提供新理论和新方法，并促进多个学科的融合和交叉，具有重要科学意义和实用价值。

非线性算子零点的分裂算法研究是非线性泛函分析与运筹学的一个交叉研究领域，它与非线性 规划、矩阵理论、凸分析和变分分析等分支有着紧密的联系，在力学、经济、交通、医学等领 域有着广泛的应用，是现代非线性分析与优化理论研究的重要课题之一。本项目在自反Banach空间框架下，对分裂算法进行了收敛性分析和应用性研究，其研究内容主要包括：(1) 借助Bregma n投影和均值技巧，设计两单调算子和算子的零点逼近算法，使之具有全局收敛性、稳定性、 快速性；(2) 运用效的数值算法求解一Ky Fan不等式的解及下半连续凸函数最小值问题。本项目为非线性算子方程求解提供新理论和新方法，在一定程度上促进了多个学科的融合和交叉，具有科学意义和实用价值。