Convergence properties of a restricted Newton-type method for generalized equations with metrically regular mappings

Abstract

Let X and Y be Banach spaces. Let $f:X\to Y$ be a Fréchet differentiable function and $F:X\rightrightarrows 2^Y$ be a set-valued mapping with closed graph. In this paper, a restricted Newton-type method has been introduced for solving the generalized equations of the form $0\in f(x)+F(x)$. Under some suitable conditions, we will establish the convergence criteria of the restricted Newton-type method, which ensures the existence and the convergence of any sequence generated by this method. More precisely, when the Fréchet derivative of f is continuous and Lipschitz continuous as well as $f+F$ is metrically regular, we analyze the semilocal and local convergence of the restricted Newton-type method. In addition, an example is given to show the reason for considering the metrically regular property instead of Lipschitz-like property of set-valued mapping in this paper.

Keywords:

• Generalized equation
• set-valued mappings
• metrically regular mappings
• restricted Newton-type method
• semi-local convergence

AMS Subject Classifications:

• 47H04
• 49J53
• 65K10

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research work of first author is fully supported by CAS-President International Fellowship Initiative(PIFI), Chinese Academy of Sciences, China and research work of the second author is partially supported by the National Natural Science Foundation of China [grant number 11331012; 11321061; 11461161005].