解非线性方程组的若干优化算法与应用研究

发布时间：2018-03-11 10:52

本文选题：非线性方程组　切入点：分式模型　出处：《南京航空航天大学》2015年博士论文　论文类型：学位论文

【摘要】：最优化理论与方法是一门应用性很强的学科,它研究如何从某些实际问题的众多可行方案中找出最优解.而非线性方程组的求解是最优化理论的重要组成部分.非线性方程组的求解在金融、贸易、管理、科学研究等国民经济的许多领域中有着广泛的应用.本论文主要是求解非线性方程组的若干优化算法与应用研究,提出了基于分式模型的解非线性方程组的信赖域法和新割线法,首次把求解非线性方程组的方法应用到求张量Z-特征值问题,提出了求对称张量Z-特征对的拟牛顿法和共轭梯度法,证明了这些算法的相关收敛性,并进行了数值计算分析,还研究了本文提出的几种算法在金融投资问题中的实际应用.整篇论文共分为八章.第一章主要介绍了本文的研究背景和意义、预备知识、研究现状和主要研究内容.第二章对求解非线性方程组的几种常用方法进行了概述.第三和第四章是求解非线性方程组的算法研究.在第三章我们提出了基于分式模型的解非线性方程组的信赖域法,证明了算法的性质,以及全局和局部二阶收敛性,并给出了该方法与信赖域牛顿法的数值对比实验结果和分析.实验结果表明对于那些曲率剧烈变化的非线性函数,分式模型逼近的效果好于一次模型.以此为基础,我们在第四章提出了基于分式模型的解非线性方程组的新割线法.该方法适用于那些雅可比矩阵难以计算的非线性方程组.在局部误差界的条件下,证明了算法的全局和局部超线性收敛性,并给出了该方法与拟牛顿法的数值对比实验结果和分析.实验结果表明新割线法对于某些问题比拟牛顿法更有效.第五和第六章是求解非线性方程组算法的应用研究,把求解对称非线性方程组的方法应用到求对称张量的Z-特征值.在第五章我们将求对称张量的特征对问题转化为求解非线性方程组的问题,证明了其非线性方程组的雅可比矩阵为对称矩阵,提出了求对称张量Z-特征对的拟牛顿法,证明了算法的全局和局部超线性收敛性,并给出了数值对比实验结果与分析.数值结果表明该方法对于求对称张量的Z-特征对在某种程度上比平移幂方法(SS-HOPM)更有效.在第六章我们将解对称非线性方程组的修正的FR共轭梯度法的线搜索技术进行改进,提出了求对称张量Z-特征对的修正的FR共轭梯度法,证明了算法的全局收敛性,并给出了数值对比实验结果与分析.实验结果表明该方法对于求对称张量的Z-特征对是有效的.第七章主要研究了本文提出的解非线性方程组的几种算法在金融投资问题中的实际应用.第八章对本文提出的算法进行了总结,并提出了一些值得进一步研究的问题.
[Abstract]:Optimization theory and method is an applied subject. It studies how to find the optimal solution from many feasible solutions of some practical problems, and the solution of nonlinear equations is an important part of optimization theory. In many fields of the national economy, such as scientific research, this paper is mainly concerned with some optimization algorithms and applications for solving nonlinear equations. A trust region method and a new Secant method for solving nonlinear equations based on fractional model are proposed. For the first time, the method of solving nonlinear equations is applied to the solution of Zhang Liang's Z- eigenvalue problem. The quasi-Newton method and conjugate gradient method for finding symmetric Zhang Liang Z- characteristic pairs are proposed. The correlation convergence of these algorithms is proved, and the numerical calculation and analysis are carried out. The thesis is divided into eight chapters. The first chapter mainly introduces the research background and significance of this paper. In the second chapter, several common methods for solving nonlinear equations are summarized. The third and 4th chapters are the algorithms for solving nonlinear equations. In chapter 3, we propose a method based on division. Trust region method for solving nonlinear equations, The properties of the algorithm and the convergence of global and local second order are proved. The numerical results and analysis of the method are compared with those of the trust region Newton method. The experimental results show that for the nonlinear functions whose curvature varies dramatically, The approximation effect of fractional model is better than that of primary model. In Chapter 4th, we propose a new Secant method for solving nonlinear equations based on fractional model. This method is suitable for nonlinear equations with Jacobian matrix which are difficult to calculate. Under the condition of local error bound, The global and local superlinear convergence of the algorithm is proved. The experimental results show that the new Secant method is more effective than the Newton method in some problems. Chapter 5th and 6th are the application research of solving nonlinear equations. The method of solving symmetric nonlinear equations is applied to find the Z- eigenvalue of symmetric Zhang Liang. In Chapter 5th, we transform the problem of finding the characteristic pairs of symmetric Zhang Liang into the problem of solving nonlinear equations. The Jacobian matrix of its nonlinear equations is proved to be a symmetric matrix, and the quasi-Newton method for finding symmetric Zhang Liang Z- characteristic pairs is presented. The global and local superlinear convergence of the algorithm is proved. The numerical results show that this method is more effective than the translation power method in finding the Z- characteristic pair of symmetric Zhang Liang. In Chapter 6th, we will solve the symmetric nonlinear equations. A modified FR conjugate gradient method for line search is improved. In this paper, a modified FR conjugate gradient method for finding symmetric Zhang Liang Z- characteristic pairs is proposed, and the global convergence of the algorithm is proved. The experimental results show that this method is effective for finding the Z- characteristic pair of symmetric Zhang Liang. Chapter 7th mainly studies several algorithms proposed in this paper for solving nonlinear equations in finance. Chapter 8th summarizes the algorithm proposed in this paper. Some problems worthy of further study are put forward.
【学位授予单位】：南京航空航天大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：O241.7

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