兰州大学机构库 >数学与统计学院
一类改进的Liu-Storey共轭梯度算法及其收敛性分析
Alternative TitleAn improved algorithm for the gradient of Liu-Storey conjugate and convergence analysis
张静也
Subtype硕士
Thesis Advisor杨爱利
2021-05-24
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name理学硕士
Degree Discipline计算数学
Keyword无约束优化 非单调线搜索 共轭梯度法 全局收敛性 修正的Liu- Storey共轭梯度法
Abstract无约束最优化问题在现实生活中有着极为广泛的应用,非线性共轭梯度算法是用来求解无约束最优化问题的一类十分重要的方法,其显著优点是所需存储量小且有较好的收敛性质,因而十分适合用于求解大规模优化问题。在具体使用中主要有Hesfenes-Stiefel(HS)法、Fletcher-Reeves(FR)法、Polak-Ribiere-Polyak(PRP)法以及Liu-Storey(LS)法等,本文主要讨论的是LS方法。 LS方法在实际计算中有着较好的数值结果,但是其收敛性却不尽如意。因此本文在前面学者的研究基础上,对LS共轭梯度法如何改进及其收敛性进行了分析和探讨。主要包括:1.通过一类改进后的非单调线搜索,使得LS共轭梯度法在迭代过程中满足充分下降性,与此同时给出了全局收敛性的证明。2.对用于LS共轭梯度算法的参数进行修正得到了一个新公式,之后以新公式为基础建立了算法框架。在不依赖于任何线搜索的条件下,证明了由新公式建立的算法框架产生的迭代方向均满足充分下降条件,并结合其他学者提出的改进弱Wolfe-Powell线搜索,得到了新的LS共轭梯度算法,并且证明了该算法的全局收敛性。最后,对新算法进行数值实验,实验结果表明了本文所改进的方法是有效的。
Other AbstractThe problem of unconstrained optimization is widely used in real life. Nonlinear conjugate gradient method is an important method to solve the problem of unconstrained optimization, which has the obvious advantage that the required storage quantity is small and has good convergence properties, so it is especially suitable for solving the problem of large-scale optimization. In the specific use of the Hesfenes-Stiefel(HS) method, Fletcher-Reeves(FR) method, Polak-Ribiere-Polyak(PRP) method and Liu-Storey(LS) method, the main discussion of this paper is the LS method. The LS method has good numerical results in the actual calculation, but its convergence is not satisfactory. Therefore, on the basis of the previous scholars'research, this paper analyzes and discusses how to improve the method of LS conjugate gradient and convergence.Mainly includes: 1. Through a kind of improved non-monotonous line search, the method of the LS conjugate gradient meets the full decline in the iteration process, and at the same time gives the proof of global convergence. 2. A new formula was obtained by correcting the parameter of the LS conjugate gradient method, and an algorithmic framework was established with the new formula. Under the condition of not relying on any line search, it is proved that the iterative directions generated by the algorithm framework established by the new formula meet the sufficient descent condition, and combined with the improved weak Wolfe-Powell line search proposed by other scholars, a new line search is obtained. LS conjugate gradient algorithm, and proves the global convergence of the algorithm. Finally, numerical experiments on the new algorithm show that the improved method is effective.
Pages37
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/460093
Collection数学与统计学院
Affiliation
数学与统计学院
First Author AffilicationSchool of Mathematics and Statistics
Recommended Citation
GB/T 7714
张静也. 一类改进的Liu-Storey共轭梯度算法及其收敛性分析[D]. 兰州. 兰州大学,2021.
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