最优化方法课程设计参考模版.doc

《最《最优优化化方方法》法》课课程程设设计计题目: 共轭梯度法算法分析与实现院系: 数学与计算科学学院专业: 数学与应用数学姓名: 梁婷艳学号: 0800730103 指导教师: 李丰兵日期: 2015 年 12月 30日摘要在各种优化算法中,共轭梯度法是非常重要的一种。本文主要介绍的共轭梯度法是介于最速下降法与牛顿法之间的一种无约束优化算法,它具有超线性收敛速度,而且算法结构简单,容易编程实现。在本次实验中, 我们首先分析共轭方向法、对该算法进行分析, 运用基于共轭方向的一种算法—共轭梯度法进行无约束优化问题的求解。无约束最优化方法的核心问题是选择搜索方向。共轭梯度法的基本思想是把共轭性与最速下降方法相结合,利用已知点处的梯度构造一组共轭方向,并沿这组方向进行搜索,求出目标函数的极小点。根据共轭方向的基本性质,这种方法具有二次终止性。再结合该算法编写 matlab 程序,求解无约束优化问题, 再结合牛顿算法的理论知识, 编写 matlab 程序,求解相同的无约束优化问题,进行比较分析,得出共轭梯度法和牛顿法的不同之处以及共轭梯度法的优缺点。共轭梯度法仅需利用一阶导数信息,避免了牛顿法需要存储和计算 Hesse 矩阵并求逆的缺点, 共轭梯度法不仅是解决大型线性方程组最有用的方法之一,也是解大型非线性最优化最有效的算法之一。共轭梯度法是一个典型的共轭方向法,它的每一个搜索方向是互相共轭的,而这些搜索方向仅仅是负梯度方向与上一次迭代的搜索方向的组合,因此,存储量少, 计算方便。关键词:共轭梯度法;超线性收敛;牛顿法; 无约束优化 Abstract Ina variety of optimization algorithms, conjugate gradient method isa very important one. In this paper, the conjugate gradient method is between the steepest descent method and Newton method for unconstrained optimization between a method, it has superlinear convergence rate, and the algorithm is simple and easy programming. In this experiment, we first analyze the conjugate direction method, the algorithm analysis, the use ofa conjugate direction-based algorithm - conjugate gradient method for unconstrained optimization problems. Unconstrained optimization method is to select the core issue of the search direction. Conjugate gradient method is the basic idea of the conjugate descent method with the bined points in the gradient using the known structure ofa set of conjugate directions, and search along the direction of this group, find the minimum point of objective function. According to the basic nature of the conjugate direction, this method has the quadratic termination. Combined with the preparation of this algorithm matlab program for solving unconstrained optimization problems, combined with Newton ’s theory of knowledge, writing matlab program to solve the same problem of unconstrained optimization, comparison analysis, the conjugate gradient method and Newton method different Office and