Inexact Newton method for non-linear functions with values in a cone

ABSTRACT

The problem of finding a solution of non-linear inclusion problems in Banach space is considered in this paper. Using convex optimization techniques introduced by Robinson (Numer Math. 1972;19:341–347), a robust convergence theorem for the inexact Newton method is proved. As an application, an affine invariant version of Kantorovich’s theorem and Smale’s $\alpha$-theorem for the inexact Newton method is obtained.

KEYWORDS:

• Inclusion problems
• inexact Newton method
• majorant condition
• semi-local convergence

AMS SUBJECT CLASSIFICATIONS:

• 65K15
• 49M15
• 90C31

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by CAPES [Projeto 019/2011]- Cooperação Internacional Brasil-China), CNPq [grant number 471815/2012-8], [grant number 444134/2014-0] and [grant number 305158/2014-7], PRONEX–Optimization (FAPERJ/CNPq) and FAPEG/GO.