Convergence Analysis of General Version of Gauss-Type Proximal Point Method for Metrically Regular Mappings

Md. Asraful Alom,Kalyan Kumer Dey,Mohammed Harunor Rashid

doi:10.4236/am.2016.711110

Abstract

We introduce and study in the present paper the general version of Gauss-type proximal point algorithm (in short GG-PPA) for solving the inclusion , where T is a set-valued mapping which is not necessarily monotone acting from a Banach space X to a subset of a Banach space Y with locally closed graph. The convergence of the GG-PPA is present here by choosing a sequence of functions with , which is Lipschitz continuous in a neighbourhood O of the origin and when T is metrically regular. More precisely, semi-local and local convergence of GG-PPA are analyzed. Moreover, we present a numerical example to validate the convergence result of GG-PPA.

Highlights

We are concerned in this study with the problem of finding a point x ∈ Ω ⊆ X satisfying 0∈T (x), (1)where T : X 2Y is a set-valued mapping and X and Y are Banach spaces
We recall the following statement of Lyusternik-Graves theorem for metrically regular mapping from [21]
In the particular case, when x is a solution of (1), that is, y = 0, Theorem 1 is reduced to the following corollary, which gives the local convergence of the sequence generated by the general version of Gauss-type proximal point algorithm (GG-PPA) defined by Algorithm

Summary

Introduction

We are concerned in this study with the problem of finding a point x ∈ Ω ⊆ X satisfying. (2016) Convergence Analysis of General Version of GaussType Proximal Point Method for Metrically Regular Mappings. From the viewpoint of numerical computation, we can assume that these kinds of methods are not suitable in practical application This drawback motivates us to introduce a method “socalled” general version of Gauss-type proximal point algorithm (GG-PPA). In [4], Rashid et al have given a semilocal convergence analysis for the classical Gauss-type proximal point method. Our aim is to study the semilocal convergence for the GG-PPA defined by Algorithm 2. In the last Section, we will give a summary of the major results to close our paper

Notations and Preliminary Results

Convergence Analysis of GG-PPA

Numerical Experiment

Conclusions