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A wonderful journey of approximate fixed points: how to find the solution with a simple algorithm?
Fixed-point computation is the process of computing the exact or approximate fixed point of a given function. This occupies an important position in mathematics, especially in game theory, economics and dynamic system analysis, and has wide applications. According to Brouwer's fixed point theorem, if a function is continuous and can map the unit d-cube onto itself, it must have a fixed point. Although the theoretical proof is not constructive, with the development of algorithms, many methods are able to calculate approximate fixed points.
“Approximate fixed-point algorithms not only improve computational efficiency, but also provide solutions in a variety of application areas, such as economic models and dynamic systems.”
In mathematics, the unit interval is often denoted by E := [0, 1], and the unit d-dimensional cube is E^d. For a continuous function f defined on E^d, the process of finding its fixed point x is to hope to achieve f(x) = x. But when faced with general functions, since the fixed points may be arbitrary real numbers, it becomes impossible to calculate the fixed points accurately. This is why the calculation algorithm of approximate fixed points is particularly important.
It is generally agreed that the standards for approximate fixed points include residual standards, absolute standards and relative standards. First, the residual criterion requires a fixed point x to satisfy |f(x) - x| ≤ ε, while the absolute criterion is |x - x₀| ≤ δ, where x₀ is some fixed point. Furthermore, there are certain interrelationships and limitations between these three criteria when considering Lipschitz continuous functions.
“For each contracting function, using the Banach fixed point iteration algorithm will greatly simplify the process of finding fixed points.”
Banach's fixed point theorem states that for a contract mapping, if a fixed point iteration method is used, the error is only in the range of O(L^t) after t iterations. This means that the number of evaluations required is logarithmic in the number of δ relative to the number of fixed points. Of course, as the Lipschitz constant L approaches 1, the number of required evaluations grows infinitely. It can be seen from this that the performance of the solution algorithm will change significantly as the parameters change.
For a one-dimensional function, using the bisection method we can find a δ-absolute fixed point within O(log(1/δ)) number of queries, which means we can re-partition the interval according to the value of the current midpoint in each iteration and eventually get the desired result. However, in higher dimensions, the challenge increases significantly, since fixed points can only be found in more complex spaces.
"In high-dimensional spaces, the number of evaluations required to find a fixed point can be infinite, especially when the exact nature of the function is unknown."
In addition to traditional iterative algorithms, various new algorithms developed by Harold Kuhn and Herbert Scarf also provide more solutions to fixed-point problems. These algorithms perform well for certain types of functions (such as Lipschitz continuous functions), and further research has enabled these traditional algorithms to be optimized, thereby improving computational efficiency.
Recent new algorithms such as BEFix and BEDFix are specifically designed to handle approximate fixed point problems of two-dimensional functions, and the efficiency of operations is greatly improved. These optimized algorithms all rely on the number of logarithmic queries, providing users with a basic operating framework to achieve higher computing speed and accuracy.
"With the development of algorithms, we can maintain stable and efficient evaluation results when calculating complex problems."
In the next development, understanding the properties of functions and continuously optimizing existing calculation methods will be the key to our further exploration of fixed points. Whether it is market equilibrium in economics or Nash equilibrium in game theory, the application of these algorithms demonstrates the close connection between mathematics and practical applications. Can we further advance these fixed-point computational algorithms in future research to unlock their greater potential in a wider range of applications?
