Approximate Solutions of Common Fixed Point Problems - Springer Optimization And

Fixed point problems occur across various domains of mathematics and play a crucial role in optimization theory. Finding approximate solutions to these problems is often a challenging task that requires careful analysis and sophisticated algorithms. In this article, we will explore the concept of approximate solutions for common fixed point problems and understand their significance in the field of optimization.

Understanding Fixed Point Problems

In mathematics, a fixed point of a function is a point that remains unchanged when the function is applied. Fixed point problems involve finding such points, and they appear in a wide range of mathematical problems, including optimization, economics, and computer science.

Fixed point problems are typically expressed as the equation:

Approximate Solutions of Common Fixed-Point Problems (Springer Optimization and Its Applications Book 112)

by Alexander J. Zaslavski(1st ed. 2016 Edition, Kindle Edition)

4.4 out of 5

Language	:	English
File size	:	5638 KB
Print length	:	463 pages
Screen Reader	:	Supported

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x = f(x)

Here, x represents the fixed point, and f(x) is the function that operates on x. The goal is to find a value for x that satisfies the equation.

The Significance of Approximate Solutions

Exact solutions to fixed point problems can sometimes be elusive or computationally expensive to obtain. In such cases, approximate solutions offer a practical alternative. Approximate solutions provide value by providing an estimate that is close enough to the true solution for practical purposes.

Approximation techniques play a crucial role in optimization theory, where finding the exact solution may not be feasible due to the complexity of the problem or limited computational resources. By finding an approximate solution, we can still make significant progress and achieve desirable outcomes.

Common Techniques for Approximate Solutions

Several techniques have been developed to approximate solutions to common fixed point problems. Let's explore some of them:

Iterative Methods

Iterative methods are widely used to find approximate solutions to fixed point problems. These methods involve repeatedly applying a function to an initial guess until a desired level of convergence is achieved. The computed sequence of values gradually moves closer to the true fixed point.

Some popular iterative methods include the Banach fixed-point theorem, Newton's method, and the bisection method.

Fixed Point Iterations

In fixed-point iterations, a modified equation is formed using the original fixed point problem equation. This modified equation, often referred to as a contraction mapping, is specially designed to ensure the existence and uniqueness of the solution. The iteration process involves repeatedly applying the contraction mapping to an initial approximation.

The widely used Banach fixed-point theorem ensures that the iteration process converges to the unique fixed point.

Optimization-Based Approaches

Fixed point problems can often be formulated as optimization problems. By representing the fixed point equation as an optimization problem, approximation techniques from the field of optimization theory can be used to find approximate solutions.

Techniques such as gradient descent, interior-point methods, and genetic algorithms can be employed to approximate the fixed point solution.

Applications of Approximate Solutions in Optimization

Approximate solutions to fixed point problems find extensive applications in optimization problems across various fields. Some notable examples include:

• Image Processing: Approximate fixed points are used in image restoration, denoising, and super-resolution applications.
• Economics and Finance: Approximate solutions help in analyzing optimal decision-making in economic models and pricing derivative securities.
• Machine Learning: Approximate fixed points are utilized in training deep neural networks and optimizing loss functions.
• Computer Vision: Approximate solutions are employed in object tracking, image registration, and 3D reconstruction.

The ability to find approximate solutions to fixed point problems opens up avenues for solving complex optimization problems efficiently.

Approximate solutions to common fixed point problems are instrumental in optimization theory and various other mathematical fields. They allow us to find practical solutions when exact solutions are challenging to obtain. By employing iterative methods, fixed point iterations, and optimization-based approaches, we can approximate fixed points efficiently and accurately, expanding the realm of solvable optimization problems. The applications of approximate solutions in fields like image processing, economics, finance, machine learning, and computer vision reinforce their significance in modern problem-solving approaches.

Approximate Solutions of Common Fixed-Point Problems (Springer Optimization and Its Applications Book 112)

by Alexander J. Zaslavski(1st ed. 2016 Edition, Kindle Edition)

4.4 out of 5

Language	:	English
File size	:	5638 KB
Print length	:	463 pages
Screen Reader	:	Supported

FREE

This book presents results on the convergence behavior of algorithms which are known as vital tools for solving convex feasibility problems and common fixed point problems. The main goal for us in dealing with a known computational error is to find what approximate solution can be obtained and how many iterates one needs to find it. According to know results, these algorithms should converge to a solution. In this exposition, these algorithms are studied, taking into account computational errors which remain consistent in practice. In this case the convergence to a solution does not take place. We show that our algorithms generate a good approximate solution if computational errors are bounded from above by a small positive constant.

Beginning  with an , this monograph moves on to study:

· dynamic string-averaging methods for common fixed point problems in a Hilbert space

· dynamic string methods for common fixed point problems in a metric space<

· dynamic string-averaging version of the proximal algorithm

· common fixed point problems in metric spaces

· common fixed point problems in the spaces with distances of the Bregman type

· a proximal algorithm for finding a common zero of a family of maximal monotone operators

· subgradient projections algorithms for convex feasibility problems in Hilbert spaces