Algorithms for Solving Common Fixed Point Problems

verfasst von: Alexander J. Zaslavski

Verlag: Springer International Publishing

Buchreihe : Springer Optimization and Its Applications

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book details approximate solutions to common fixed point problems and convex feasibility problems in the presence of perturbations. Convex feasibility problems search for a common point of a finite collection of subsets in a Hilbert space; common fixed point problems pursue a common fixed point of a finite collection of self-mappings in a Hilbert space. A variety of algorithms are considered in this book for solving both types of problems, the study of which has fueled a rapidly growing area of research. This monograph is timely and highlights the numerous applications to engineering, computed tomography, and radiation therapy planning.

Totaling eight chapters, this book begins with an introduction to foundational material and moves on to examine iterative methods in metric spaces. The dynamic string-averaging methods for common fixed point problems in normed space are analyzed in Chapter 3. Dynamic string methods, for common fixed point problems in a metric space are introduced and discussed in Chapter 4. Chapter 5 is devoted to the convergence of an abstract version of the algorithm which has been called component-averaged row projections (CARP). Chapter 6 studies a proximal algorithm for finding a common zero of a family of maximal monotone operators. Chapter 7 extends the results of Chapter 6 for a dynamic string-averaging version of the proximal algorithm. In Chapters 8 subgradient projections algorithms for convex feasibility problems are examined for infinite dimensional Hilbert spaces.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

In this book we study approximate solutions of common fixed point and convex feasibility problems in the presence of perturbations. A convex feasibility problem is to find a point which belongs to the intersection of a given finite family of convex subsets of a Hilbert space. This problem is a special case of a common fixed point problem which is to find a common fixed point of a finite family of nonlinear mappings in a Hilbert space. Our goal is to show the convergence of algorithms, which are known as important tools for solving convex feasibility and common fixed point problems. Some of these algorithms are discussed is this chapter.

Alexander J. Zaslavski

Chapter 2. Iterative Methods in Metric Spaces

Abstract

In this chapter we study the convergence of iterative methods for solving common fixed point problems in a metric space. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact iterative method generates an approximate solution if perturbations are summable. We also show that if the mappings are nonexpansive and the perturbations are sufficiently small, then the inexact method produces approximate solutions.

Alexander J. Zaslavski

Chapter 3. Dynamic String-Averaging Methods in Normed Spaces

Abstract

In this chapter we study the convergence of dynamic string-averaging methods for solving common fixed point problems in a normed space. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact dynamic string-averaging algorithm generates an approximate solution if perturbations are summable. We also show that if the mappings are nonexpansive and the perturbations are sufficiently small, then the inexact method produces approximate solutions.

Alexander J. Zaslavski

Chapter 4. Dynamic String-Maximum Methods in Metric Spaces

Abstract

In this chapter we study the convergence of dynamic string-maximum methods for solving common fixed point problems in a metric space. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact iterative method generates an approximate solution if perturbations are summable.

Alexander J. Zaslavski

Chapter 5. Abstract Version of CARP Algorithm

Abstract

In this chapter we study the convergence of an abstract version of the algorithm which is called in the literature as component-averaged row projections or CARP. This algorithm was introduced for solving a convex feasibility problem in a finite-dimensional space, when a given collection of sets is divided into blocks in such a manner that all sets belonging to every block are subsets of a vector subspace associated with the block. All the blocks are processed in parallel and the algorithm operates in vector subspaces of the whole vector space. This method becomes efficient, in particular, when the dimensions of the subspaces are essentially smaller than the dimension of the whole space. In the chapter we study CARP for problems in a normed space, which is not necessarily finite-dimensional. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact dynamic string-averaging algorithm generates an approximate solution if perturbations are summable. We also show that if the mappings are nonexpansive and the perturbations are sufficiently small, then the inexact dynamic string-averaging algorithm produces approximate solutions.

Alexander J. Zaslavski

Chapter 6. Proximal Point Algorithm

Abstract

In a Hilbert space, we study the convergence of an iterative proximal point method to a common zero of a finite family of maximal monotone operators under the presence of perturbations. We show that the inexact proximal point method generates an approximate solution if perturbations are summable. We also show that if the perturbations are sufficiently small, then the inexact proximal point method produces approximate solutions.

Alexander J. Zaslavski

Chapter 7. Dynamic String-Averaging Proximal Point Algorithm

Abstract

In a Hilbert space, we study the convergence of a dynamic string-averaging proximal point method to a common zero of a finite family of maximal monotone operators under the presence of perturbations. Our main goal is to obtain an approximate solution of the problem using perturbed algorithms. We show that the inexact dynamic string-averaging proximal point algorithm generates an approximate solution if perturbations are summable. We also show that if the perturbations are sufficiently small, then the inexact produces approximate solutions.

Alexander J. Zaslavski

Chapter 8. Convex Feasibility Problems

Abstract

We use inexact subgradient projection algorithms for solving convex feasibility problems. We show that almost all iterates, generated by a perturbed subgradient projection algorithm in a Hilbert space, are approximate solutions. Moreover, we obtain an estimate of the number of iterates which are not approximate solutions.

Alexander J. Zaslavski

Backmatter

Titel: Algorithms for Solving Common Fixed Point Problems
verfasst von: Alexander J. Zaslavski
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-77437-4
Print ISBN: 978-3-319-77436-7
DOI: https://doi.org/10.1007/978-3-319-77437-4