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Zbl 0776.65037
Qi, Liqun
Convergence analysis of some algorithms for solving nonsmooth equations. (English)
[J] Math. Oper. Res. 18, No.1, 227-244 (1993). ISSN 1526-5471; ISSN 0364-765X/e

For the solution of nonlinear systems $F(x)=0$, where $F: \bbfR\sp n\to \bbfR\sp n$ is locally Lipschitzian and directionally differentiable, some modifications of Newton's method have been based on directional derivatives [see {\it J. S. Pang}, Math. Program. Ser. A 51, No. 1, 101- 131 (1991; Zbl 0733.90063)] or on the use of generalized Jacobians of $F$ in the sense of {\it F. H. Clarke} [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)].\par Here a convergence analysis of these two approaches is presented. Local superconvergence is proved under certain regularity conditions that are the nonsmooth analogue of the nonsingularity of the derivative in the smooth case. Global convergence of the damped, directional-derivative form of Newton's method is studied.\par Finally a general attraction theorem is proved that applies, for example, to the two algorithms considered by {\it S. P. Han}, {\it J. S. Pang} and {\it N. Rangaraj} [Math. Oper. Res. 17, No. 3, 586-607 (1992)], as well as to a new hybrid method given here.
[W.C.Rheinboldt (Pittsburgh)]

MSC 2000:: *65H10 Systems of nonlinear equations (numerical methods)

Keywords: local convergence; local superconvergence; global convergence; nonlinear systems; Newton's method

Citations: Zbl 0733.90063; Zbl 0582.49001

Cited in: Zbl 1236.90146 Zbl 0993.49017

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