大规模非线性方程组问题的有限记忆拟牛顿方法研究

项目名称： 大规模非线性方程组问题的有限记忆拟牛顿方法研究

项目编号： No.11261006

项目类型： 地区科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 袁功林

作者单位： 广西大学

项目金额： 50万元

中文摘要： 非线性方程组的数值方法有信赖域法、牛顿法和拟牛顿法等，这些方法需计算和存储雅克比矩阵或校正矩阵，占用大量计算机内存，很难求解其大规模问题。有限记忆拟牛顿方法是近年来出现的求解无约束优化问题非常有效的算法，它具有收敛速度快、存储量小的优点，特别适合大规模问题。我们将此方法应用于非线性方程组问题，得到的初步成果已发表在SCI期刊上(MCM,54(2011))，可有限记忆拟牛顿方法在非线性方程组中还没有得到广泛应用。如何发挥有限记忆拟牛顿方法低存储、速度快的优点，并成功求解大规模非线性方程组问题，是很有意义的研究工作。 鉴于此，本项目具体研究：1、大规模非线性方程组的有限记忆拟牛顿算法研究，获得全局收敛性，将其与信赖域方法结合，得到高阶收敛速度；2、采用超松弛技术，设计出快速的有限记忆拟牛顿算法；3、数值结果方面，利用Fortran（或Matlab）语言成功求解至少10万维的非线性方程组问题。

中文关键词： 大规模；非线性方程组；拟牛顿方法；收敛性；非光滑优化

英文摘要： There are many numerical methods which are the trust region method, Newton method, and quasi-Newton method etc. to solve nonlinear equations. The computation and storage of the Jacobian matrix (or a correction matrix) is necessary, which occupies obviously a large number of computer memory. Then it is difficult to solve large scale problems. The limited memory quasi-newton method is one of the most effective methods for unconstrained optimization problems, which has fast convergent rate and less storage space. Moreover, it is suitable for solving large-scale problems. This method is used to solve nonlinear equations and the paper is published in journal of SCI (MCM, 54(2011)). However, it has not been widely used in nonlinear equations problems. How to play the low storage and quick speed of the limited memory quasi-Newton method and solve successfully large-scale nonlinear equations problems is a significance work. In view of the above considerations, we will make the following studies: 1.The limited memory quasi-newton algorithms for large-scale nonlinear equations are studied, which have the global convergence and the high-order convergence rate combining with the trust region method; 2.The super relaxation technique is used and the rapid limited memory quasi-newton algorithms are designed; 3.For numerical re

英文关键词： large scale；nonlinear equations；quasi-Newton method；convergence；nonsmooth optimization