On Modelling and Complete Solutions to General Fixpoint Problems in Multi-Scale Systems with Applications

TL;DR

This paper presents a unified approach using canonical duality theory to model and solve fixed point problems in multi-scale systems, providing comprehensive solutions and stability analysis for complex nonconvex problems.

Contribution

It reformulates fixed point problems as nonconvex optimization problems and applies canonical duality theory to find all fixed points and analyze their stability.

Findings

Successfully solves fixed point problems with nonconvex operators

Provides stability properties of solutions within the duality framework

Demonstrates the approach on polynomial, exponential, and logarithmic cases

Abstract

This paper revisits the well-studied fixed point problem from a unified viewpoint of mathematical modeling and canonical duality theory, i.e. the original problem is first reformulated as a nonconvex optimization problem, its well-posedness is discussed based on objectivity principle in continuum physics; then the canonical duality theory is applied for solving this problem to obtain not only all fixed points, but also their stability properties. Applications are illustrated by challenging problems governed by nonconvex polynomial, exponential, and logarithmic operators. This paper shows that within the framework of the canonical duality theory, there is no difference between the fixed point problems and nonconvex analysis/optimization in multidisciplinary studies.