Convergence analysis of some algorithms for solving nonsmooth equations. (English) Zbl 0776.65037

Qi, Liqun

Math. Oper. Res. 18, No. 1, 227-244 (1993).

For the solution of nonlinear systems \(F(x)=0\), where \(F: \mathbb{R}^ n\to \mathbb{R}^ n\) is locally Lipschitzian and directionally differentiable, some modifications of Newton’s method have been based on directional derivatives [see 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 F. H. Clarke [Optimization and nonsmooth analysis (1983; Zbl 0582.49001)].
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.
Finally a general attraction theorem is proved that applies, for example, to the two algorithms considered by S. P. Han, J. S. Pang and N. Rangaraj [Math. Oper. Res. 17, No. 3, 586-607 (1992)], as well as to a new hybrid method given here.

Reviewer: W.C.Rheinboldt (Pittsburgh)

MSC:

65H10

Numerical computation of solutions to systems of equations

Keywords:

local convergence; local superconvergence; global convergence; nonlinear systems; Newton’s method

Citations:

Zbl 0733.90063; Zbl 0582.49001