Descent directions of quasi-Newton methods for symmetric nonlinear equations
Descent directions of quasi-Newton methods for symmetric nonlinear equations
复制标题
DOI：
10.1137/s0036142901397423
复制
发表时间：
2002-12-03
影响因子：
2.9
通讯作者：
Zhou, SZ
中科院分区：
文献类型：
--
作者：
Gu, GZ;Li, DH;Zhou, SZ
一般来说，当应用拟牛顿法求解非线性方程组时，拟牛顿方向不一定是范数函数的下降方向。在本文中，我们表明，当应用于求解对称非线性方程时，具有正定迭代矩阵的拟牛顿法可以生成范数函数的下降方向。基于高斯牛顿 BFGS 方法 [D. H. Li 和 M. Fukushima，SIAM J. Numer。 Anal., 37 (1999), pp. 152-172]，我们开发了一种范数下降 BFGS 方法来求解对称非线性方程。在温和的条件下，我们建立了该方法的全局和超线性收敛性。所提出的方法具有 BFGS 方法在解决无约束优化问题时的一些有利特性： (a) 生成的拟牛顿矩阵序列是正定的； (b) 生成的迭代序列是范数下降； (c) 在没有雅可比行列式非奇异性假设的情况下，建立了全局收敛定理。报告的初步数值结果积极支持了该方法。
In general, when a quasi-Newton method is applied to solve a system of nonlinear equations, the quasi-Newton direction is not necessarily a descent direction for the norm function. In this paper, we show that when applied to solve symmetric nonlinear equations, a quasi-Newton method with positive definite iterative matrices may generate descent directions for the norm function. On the basis of a Gauss Newton based BFGS method [D. H. Li and M. Fukushima, SIAM J. Numer. Anal., 37 (1999), pp. 152-172], we develop a norm descent BFGS method for solving symmetric nonlinear equations. Under mild conditions, we establish the global and superlinear convergence of the method. The proposed method shares some favorable properties of the BFGS method for solving unconstrained optimization problems: (a) the generated sequence of the quasi-Newton matrices is positive definite; (b) the generated sequence of iterates is norm descent; (c) a global convergence theorem is established without nonsingularity assumption on the Jacobian. Preliminary numerical results are reported, which positively support the method.