A Self-Adaptive Algorithm for the Common Solution of the Split Minimization Problem and the Fixed Point Problem

Nattakarn Kaewyong,Kanokwan Sitthithakerngkiet

doi:10.3390/axioms10020109

Nattakarn Kaewyong, Kanokwan Sitthithakerngkiet

Open Access

https://doi.org/10.3390/axioms10020109

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Journal: Axioms	Publication Date: May 30, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: King Mongkut's University of Technology North Bangkok

Abstract

In this paper, a new self-adaptive step size algorithm to approximate the solution of the split minimization problem and the fixed point problem of nonexpansive mappings was constructed, which combined the proximal algorithm and a modified Mann’s iterative method with the inertial extrapolation. The strong convergence theorem was provided in the framework of Hilbert spaces and then proven under some suitable conditions. Our result improved related results in the literature. Moreover, some numerical experiments were also provided to show our algorithm’s consistency, accuracy, and performance compared to the existing algorithms in the literature.

Highlights

Throughout this paper, we denote two nonempty closed convex subsets of two realHilbert spaces H1 and H2 by C and Q, respectively
Based on the above ideas, the aims of this work were: (1) to construct a new selfadaptive step size algorithm combine with the proximal algorithm, the modified Mann method with the inertial extrapolation to solve the split minimization problem (SMP)
(10), and the fixed point problems of a nonexpansive mapping; (2) to establish the strong convergence results for the SMP and fixed point problems using the proposed algorithm; (3) to give numerical examples for our algorithm to present its consistency, accuracy, and performance compared to the existing algorithms in the literature

Summary

Introduction

Throughout this paper, we denote two nonempty closed convex subsets of two realHilbert spaces H1 and H2 by C and Q, respectively. Throughout this paper, we denote two nonempty closed convex subsets of two real. We denote the orthogonal projections onto a set C by PC and let A∗ : H2 → H1 be an adjoint operator of A : H1 → H2 , where A is a bounded linear operator. Inverse problems have been widely studied since they stand at the core of image reconstruction problems and signal processing. The split feasibility problem (SFP) is one of the most popular inverse problems that has attracted the attention of many researchers. Cencer and Elfving first considered the split feasibility problem (SFP). The split feasibility problem (SFP) can mathematically be expressed as follows: find an element x with: Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in x ∈ C such that Ax ∈ Q

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Results

Conclusion