基于最小二乘的三项共轭梯度方法

发布时间：2018-04-24 12:59

本文选题：无约束优化 + 非线性方程组　；参考：《广西大学》2017年硕士论文

【摘要】：最优化作为运筹学与控制论学科的一个重要组成部分,其研究的问题广泛来源于实际应用,比如常见的有经济管理、工程设计、最优控制、石油勘探等问题.无约束优化问题是优化领域研究的一类基本而重要的问题.求解无约束最优化问题的方法主要有最速下降法、牛顿法、拟牛顿法、共轭梯度法和信赖域方法等.其中,共轭梯度法迭代简单,存储和计算量小,是求解大规模无约束优化问题较有效的方法之一.目前三项共轭梯度法是共轭梯度方法研究的热点之一,本文主要研究求解无约束最优化问题和非线性方程组的三项共轭梯度方法.针对无约束优化问题,本文基于最小二乘技术提出了求解大规模无约束最优化问题的三项共轭梯度方法.对数值结果和性质比较好的三项共轭梯度方法,结合最小二乘技术对其进行逼近,提出新的三项共轭梯度法迭代公式.该算法具有如下优点:(1)算法在不考虑线搜索的前提下,具有下降性,即算法的下降性不依赖于线搜索技术的选择;(2)在一定的条件下,算法具有全局收敛性;(3)通过数值试验,说明算法对于大规模无约束优化问题具有很好的数值结果.对大规模非线性方程组问题进行研究,提出改进的Polak-Ribiere-Polyak(PRP)投影三项共轭算法,并证明该算法的全局收敛性,由于本文提出的算法具有低存储的优点,因而可以用来求解大规模非线性方程组.数值结果表明当方程组的维数较高时,该算法仍具有很好的数值结果。
[Abstract]:Optimization is an important part of operational research and cybernetics. Its research problems come from practical applications, such as economic management, engineering design, optimal control, petroleum exploration and so on. Unconstrained optimization problem is a kind of basic and important problem in optimization field. The main methods for solving unconstrained optimization problems include the steepest descent method, Newton method, quasi-Newton method, conjugate gradient method and trust region method. The conjugate gradient method is one of the most effective methods for solving large scale unconstrained optimization problems because of its simple iteration and small amount of storage and computation. At present, three term conjugate gradient method is one of the hot topics in the study of conjugate gradient method. In this paper, we mainly study three conjugate gradient methods for solving unconstrained optimization problems and nonlinear equations. For unconstrained optimization problems, this paper presents three conjugate gradient methods for solving large-scale unconstrained optimization problems based on least square technique. For the three term conjugate gradient method which has good numerical results and good properties, a new iterative formula of the three term conjugate gradient method is proposed by combining the least square technique. The algorithm has the following advantages: (1) the algorithm has the descending property without considering the line search, that is, the descent of the algorithm does not depend on the selection of the line search technology.) under certain conditions, the algorithm has global convergence. It is shown that the algorithm has good numerical results for large scale unconstrained optimization problems. In this paper, a modified Polak-Ribiere-Polyak-PRP (Polak-Ribiere-Polyak-PRP) projection algorithm is proposed, and the global convergence of the algorithm is proved. Therefore, it can be used to solve large scale nonlinear equations. The numerical results show that the algorithm still has good numerical results when the dimension of the equations is high.
【学位授予单位】：广西大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O224

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