Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations
Convergence Analysis of Some Algorithms for Solving Nonsmooth Equations
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DOI：
10.1287/moor.18.1.227
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发表时间：
1993-02
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影响因子：
--
通讯作者：
L. Qi
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文献类型：
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作者：
L. Qi
本文给出了求解由局部Lipschitzian函数定义的非线性方程组的几种算法的收敛分析。对于基于方向导数的牛顿方法和基于广义雅可比的牛顿方法，迭代和相应的函数值都是局部超线性收敛的。全局地，当且仅当迭代序列收敛于该点且步长最终变为1时，如果系统在该点是强BD正则且半光滑的，则由基于衰减的方向导数的牛顿法产生的迭代序列的极限点是系统的零点。在这种情况下，收敛是超线性的。给出了一个通用的吸引性定理，它可以应用于han，Pang和Rangaraj提出的两个算法。给出了一种混合方法，该方法在寻找系统范数函数的稳定点的意义下是全局收敛的，同时也是局部二次收敛的。
This paper presents convergence analysis of some algorithms for solving systems of nonlinear equations defined by locally Lipschitzian functions. For the directional derivative-based and the generalized Jacobian-based Newton methods, both the iterates and the corresponding function values are locally, superlinearly convergent. Globally, a limiting point of the iterate sequence generated by the damped, directional derivative-based Newton method is a zero of the system if and only if the iterate sequence converges to this point and the stepsize eventually becomes one, provided that the system is strongly BD-regular and semismooth at this point. In this case, the convergence is superlinear. A general attraction theorem is presented, which can be applied to two algorithms proposed by Han, Pang and Rangaraj. A hybrid method, which is both globally convergent in the sense of finding a stationary point of the norm function of the system and locally quadratically convergent, is also presented.