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Metrically regular mappings and its application to convergence analysis of a confined Newton-type method for nonsmooth generalized equations
Authors:Rashid  Mohammed Harunor  Yuan   Ya-xiang
Affiliation:1.Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China
;2.Department of Mathematics, Faculty of Science, University of Rajshahi, Rajshahi, 6205, Bangladesh
;
Abstract:Notion of metrically regular property and certain types of point-based approximations are used for solving the nonsmooth generalized equation f(x)+F(x)?0,where X and Y are Banach spaces,and U is an open subset of X,f:U→Y is a nonsmooth function and F:X■Y is a set-valued mapping with closed graph.We introduce a confined Newton-type method for solving the above nonsmooth generalized equation and analyze the semilocal and local convergence of this method.Specifically,under the point-based approximation of f on U and metrically regular property of f+F,we present quadratic rate of convergence of this method.Furthermore,superlinear rate of convergence of this method is provided under the conditions that f admits p-point-based approximation on U and f+F is metrically regular.An example of nonsmooth functions that have p-point-based approximation is given.Moreover,a numerical experiment is given which illustrates the theoretical result.
Keywords:set-valued mappings  generalized equations  metrically regular mapping  semilocal convergence  point-based approximation
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