Lect4: Conjugate gradient methods

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于 2017-05-16 11:43:00 发布

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原文链接：http://www.cnblogs.com/cihui/p/6860582.html

本文介绍了线性和非线性共轭梯度方法的基本概念及其在求解大型线性方程组和非线性优化问题中的应用。文章讨论了该方法的关键特性，包括无需矩阵存储及比最速下降法更快的收敛速度。文中还对比了几种不同的优化方法，如最速下降法、坐标下降法、牛顿法等。

1. Introduction:

Linear conjugate gradient method: solve a large linear system of equations
Nonlinear conjugate gradient method: adaption of linear CG for nonlinear optimization
Key features: requires no matrix storage, faster than steepest descent
Assume \(A\) is an \(n\times n\) symmetric pd matrix, \(\phi(x)=\frac{1}{2}x^TAx-b^Tx\) and \(\nabla\phi(x)=Ax-b=r(x)\) residual
Idea: steepest descent (1 to \(\infty\) iterations), coordinate descent (n or \(\infty\) iterations), Newton's method (1 iteration), CG(n iterations)

转载于:https://www.cnblogs.com/cihui/p/6860582.html