Fixed Point Optimization Algorithms And Their Applications
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Fixed Point Optimization Algorithms and Their Applications
Author: Watcharaporn Cholamjiak
language: en
Publisher: Morgan Kaufmann
Release Date: 2024-11-23
Fixed Point Optimization Algorithms and Their Applications discusses how the relationship between fixed point algorithms and optimization problems is connected and demonstrates hands-on applications of the algorithms in fields such as image restoration, signal recovery, and machine learning. The book is divided into nine chapters beginning with foundational concepts of normed linear spaces, Banach spaces, and Hilbert spaces, along with nonlinear operators and useful lemmas and theorems for proving the book's main results. The author presents algorithms for nonexpansive and generalized nonexpansive mappings in Hilbert space, and presents solutions to many optimization problems across a range of scientific research and real-world applications. From foundational concepts, the book proceeds to present a variety of optimization algorithms, including fixed point theories, convergence theorems, variational inequality problems, minimization problems, split feasibility problems, variational inclusion problems, and equilibrium problems. Fixed Point Optimization Algorithms and Their Applications equips readers with the theoretical mathematics background and necessary tools to tackle challenging optimization problems involving a range of algebraic methods, empowering them to apply these techniques in their research, professional work, or academic pursuits. - Demonstrates how to create hybrid algorithms for many optimization problems with non-expansive mappings to solve real-world problems - Shows readers how to solve image restoration problems using optimization algorithms - Includes coverage of signal recovery problems using optimization algorithms - Shows readers how to solve data classification problems using optimization algorithms in machine learning with many types of datasets, such as those used in medicine, mathematics, computer science, and engineering
Algorithms for Solving Common Fixed Point Problems
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.
Fixed-Point Algorithms for Inverse Problems in Science and Engineering
Author: Heinz H. Bauschke
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-05-27
"Fixed-Point Algorithms for Inverse Problems in Science and Engineering" presents some of the most recent work from top-notch researchers studying projection and other first-order fixed-point algorithms in several areas of mathematics and the applied sciences. The material presented provides a survey of the state-of-the-art theory and practice in fixed-point algorithms, identifying emerging problems driven by applications, and discussing new approaches for solving these problems. This book incorporates diverse perspectives from broad-ranging areas of research including, variational analysis, numerical linear algebra, biotechnology, materials science, computational solid-state physics, and chemistry. Topics presented include: Theory of Fixed-point algorithms: convex analysis, convex optimization, subdifferential calculus, nonsmooth analysis, proximal point methods, projection methods, resolvent and related fixed-point theoretic methods, and monotone operator theory. Numerical analysis of fixed-point algorithms: choice of step lengths, of weights, of blocks for block-iterative and parallel methods, and of relaxation parameters; regularization of ill-posed problems; numerical comparison of various methods. Areas of Applications: engineering (image and signal reconstruction and decompression problems), computer tomography and radiation treatment planning (convex feasibility problems), astronomy (adaptive optics), crystallography (molecular structure reconstruction), computational chemistry (molecular structure simulation) and other areas. Because of the variety of applications presented, this book can easily serve as a basis for new and innovated research and collaboration.