zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Convergence analysis of some algorithms for solving nonsmooth equations. (English) Zbl 0776.65037

For the solution of nonlinear systems F(x)=0, where F: n 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.

65H10Systems of nonlinear equations (numerical methods)