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Iterative methods for finding fixed points of non-expansive operators in Hilbert spaces have been described in many publications. In this monograph we try to present the methods in a consolidated way. We introduce several classes of operators, examine their properties, define iterative methods generated by operators from these classes and present general convergence theorems. On this basis we discuss the conditions under which particular methods converge. A large part of the results presented in this monograph can be found in various forms in the literature (although several results presented here are new). We have tried, however, to show that the convergence of a large class of iteration methods follows from general properties of some classes of operators and from some general convergence theorems.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

In Chap. 1 we present the notation, definitions and basic facts about convex subsets and convex functions defined on a Hilbert space, convergence and differentiation properties, the properties of matrices, etc., which will be used in further parts of the book. Then we define the metric projection for a Hilbert space and present its basic properties. This operator plays an important role in further parts of the book. Finally, we present several convex optimization problems: convex minimization, variational inequalities, convex feasibility problems and split feasibility problems. These problems as well as the methods for solving them have applications in various areas of mathematics. Furthermore, these abstract problems can be treated as mathematical models for many practical problems which arise in physical, medical, technical and information sciences.

Andrzej Cegielski

Chapter 2. Algorithmic Operators

In Chap. 2 we introduce several classes of operators: nonexpansive, quasi-nonexpansive, strictly quasi-nonexpansive, strongly quasi-nonexpansive, cutters, firmly nonexpansive, relaxed firmly nonexpansive and strongly nonexpansive. We present general properties of these operators, prove the closedness of these classes under some algebraic operations and present the properties of the subsets of fixed points of operators from these classes. The importance of these operators follows from the fact that they define algorithms for solving convex optimization problems. In one iteration of the algorithm an appropriate operator (called an algorithmic operator) defines an actualization of the current approximation of a solution of the convex optimization problem.

Andrzej Cegielski

Chapter 3. Convergence of Iterative Methods

Many convex optimization problems can be reduced to finding a fixed point of a nonexpansive (or a more general operator) $$U : \mathcal{H}\rightarrow \mathcal{H}$$ . Iterative methods for solving these problems have the form of a recurrence $${x}^{k+1} = {U}_{k}{x}^{k}$$ , $$k \geq 0$$ , where $${x}^{0} \in \mathcal{H}$$ is arbitrary, $$\{{U{}_{k}\}}_{k=0}^{\infty }$$ is a sequence of algorithmic operators $${U}_{k} : \mathcal{H}\rightarrow \mathcal{H}$$ such that $${\bigcap \limits }_{k=0}^{\infty }\mathrm{Fix}\,{U}_{k} \supseteq \mathrm{ Fix}\,U$$ . In Chap. 3 we present several convergence results for sequences generated by the above recurrence, where the operators $${U}_{k}$$ are nonexpansive, quasi-nonexpansive, strongly quasi-nonexpansive, averaged, relaxed firmly nonexpansive or strongly nonexpansive. Opial’s Theorem and its generalizations, which give sufficient conditions for the weak convergence play an important role here. We also present several results for the strong convergence.

Andrzej Cegielski

Chapter 4. Algorithmic Projection Operators

In Chap. 4 we give examples of algorithmic projection operators and show their properties. These properties are, in most cases, corollaries of general properties of operators presented in Chap. 2. Since the metric projection plays an important role in the construction of algorithmic projection operators, we give the formulas for the metric projection onto simple closed convex subsets usually used in applications. Furthermore, we give properties of a subgradient projection, an alternating projection and its generalized relaxation, a simultaneous projection, a cyclic projection and its extrapolation, an averaged alternating reflection, a Landweber operator and its projected version, a simultaneous cutter and its extrapolation and a surrogate projection.

Andrzej Cegielski

Chapter 5. Projection Methods

In Chap. 5 we present iterative methods for solving several convex optimization problems in a Hilbert space: the common fixed point problem, convex feasibility problem, split feasibility problem, variational inequality. All these problems can be reduced to finding fixed points of quasi-nonexpansive operators. We present several versions of the alternating projection method, projected gradient method, simultaneous projection method, cyclic projection methods, successive projection methods, Landweber method and its projected version, simultaneous and successive projection methods with an application of quasi-nonexpansive operators having a common fixed point, as well as extrapolated versions of the methods. The convergence of the methods discussed herein follows from the general convergence theorems presented in Chap. 3.

Andrzej Cegielski

Backmatter

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