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

Extended Newton-type Method for Nonsmooth Generalized Equation under n , α -point-based Approximation

“…In [8,Proposition 4.1], Rashid proved that for any function f admitting a PBA on a nonempty closed convex subset C of a Hilbert space H, the normal map associated with f admits a PBA on H. In our study we will show that the same result holds when we replace the normal maps f C + F in lieu of the normal maps f C . Rashid [8,14] reformulate the normal maps f C + F by simple modification of the definition of normal maps given by Robinson [13]. In [8,14] Rashid assumed the concept of point-based approximation and p-point-based approximation.…”

“…Rashid [8,14] reformulate the normal maps f C + F by simple modification of the definition of normal maps given by Robinson [13]. In [8,14] Rashid assumed the concept of point-based approximation and p-point-based approximation. Here we extend that concept to (n, α)-point-based approximation which is reformulated by Rashid [8,14], then we show that if f have a (n, α)-point-based approximation, then one can easily be constructed a (n, α)-point-based approximation for f C + F.…”

“…We are now able to construct a (n, α)-point-based approximation for the normal map f C + F provided that a (n, α)-point-based approximation exists for f. (e following proposition can be extracted from [14,Proposition 4.3]. Proposition 1.…”

“…In other words, if f doesn't possess Fréchet derivatives, it is not so clear how a Newton algorithm should be designed. (ere are many investigators have worked on this question and the applicants have presented different methods for a few things that are important in certain cases and have proved their justification; see for example [4,[8][9][10][11][12][13][14]. Several papers have worked on the Newton-type methods for nonsmooth equations and variational inequalities; see for example [8,9,15] for inspiration and advanced works on these topics.…”