非线性方程组的几类数值优化方法研究

发布时间：2018-07-05 17:46

本文选题：非线性方程组 + 有限记忆BFGS　；参考：《广西大学》2016年硕士论文

【摘要】：非线性优化是运筹学的一个重要分支,非线性优化问题来源于科学与工程计算中的许多领域,如气象预测问题、非线性有限元问题、非线性断裂问题、石油地质勘探问题等都存在着大量的大规模的非线性优化问题.非线性方程组的求解又和非线性优化问题的求解紧密相关,因而基于非线性优化问题的算法,研究非线性方程组的数值解,具有理论意义和实用价值.常见的求解非线性优化问题的算法有牛顿法、拟牛顿法、共轭梯度法、信赖域算法等.本文对有限记忆BFGS算法、非精确回溯下降算法和投影共轭梯度算法进行研究,取得的主要成果如下：(1)对有限记忆BFGS算法求解非线性方程组问题作了研究,提出了一个新的有限记忆BFGS算法,然后重点分析了它的下降性和全局收敛性等理论性质,最后设计了一些算例进行数值试验.数值结果表明新算法能够较好的求解大规模非线性方程组问题,且比BFGS算法的数值表现要好.(2)给出了一个修正的搜索方向,利用回溯线搜索技术提出了新的算法,然后证明了该算法的下降性和全局收敛性等理论性质,最后给出了新算法和BFGS算法的数值表现.数值结果表明新算法比BFGS算法的数值表现要好.(3)提出了一个新的投影共轭梯度算法,接着分析了该算法的充分下降性、信赖域性质和全局收敛性等性质,最后设计了一些算例进行数值试验.数值结果表明新算法能够较好的求解大规模非线性方程组,且比通常情况下数值结果较好的PRP算法的数值表现要好.
[Abstract]:Nonlinear optimization is an important branch of operational research. Nonlinear optimization problems are derived from many fields in science and engineering, such as weather prediction problem, nonlinear finite element problem, nonlinear fracture problem. There are a large number of nonlinear optimization problems in petroleum geological exploration. The solution of nonlinear equations is closely related to the solution of nonlinear optimization problems. Therefore, it is of theoretical significance and practical value to study the numerical solutions of nonlinear equations based on the algorithm of nonlinear optimization problems. Common algorithms for solving nonlinear optimization problems include Newton method, quasi-Newton method, conjugate gradient method, trust region algorithm and so on. In this paper, the finite memory BFGs algorithm, the inexact backtracking descent algorithm and the projection conjugate gradient algorithm are studied. The main results are as follows: (1) the finite memory BFGs algorithm is studied for solving nonlinear equations. In this paper, a new finite memory BFGS algorithm is proposed, and its theoretical properties such as descent and global convergence are analyzed. Finally, some numerical examples are designed for numerical experiments. Numerical results show that the new algorithm can solve large scale nonlinear equations better than BFGs algorithm. (2) A modified search direction is given, and a new algorithm is proposed by using backtracking search technique. Then the theoretical properties of the algorithm such as descent and global convergence are proved. Finally, the numerical representations of the new algorithm and the BFGS algorithm are given. Numerical results show that the new algorithm performs better than the BFGS algorithm. (3) A new projection conjugate gradient algorithm is proposed, and the sufficient descent, trust region and global convergence of the algorithm are analyzed. Finally, some numerical examples are designed to carry out numerical tests. Numerical results show that the new algorithm can solve large scale nonlinear equations, and the numerical performance of PRP algorithm is better than that of PRP algorithm.
【学位授予单位】：广西大学
【学位级别】：硕士
【学位授予年份】：2016
【分类号】：O224

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