Metrically regular mappings and its application to convergence analysis of a confined Newton-type method for nonsmooth generalized equations  

作　　者：Mohammed Harunor Rashid Ya-xiang Yuan 

机构地区：[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 6205,Bangladesh

出　　处：《Science China Mathematics》2020年第1期39-60,共22页中国科学：数学（英文版）

基　　金：supported by CAS-President International Fellowship Initiative (PIFI), Chinese Academy of Sciences, Beijing, China;supported by National Natural Science Foundation of China (Grants Nos. 11688101 and 11331012)

摘　　要：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.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.

关 键 词：set-valued mappings generalized equations metrically regular mapping semilocal convergence point-based approximation 

分 类 号：O175[理学—数学]