Convergence properties of a restricted Newton-type method for generalized equations with metrically regular mappings. Issue 1 (2nd January 2022) - Rashid, M. H.; Yuan, Ya-xiang
Abstract: Let X and Y be Banach spaces. Let f : X → Y be a Fréchet differentiable function and F : X ⇉ 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 ∈ 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.