Canonical Duality-Triality Theory for Solving General Global Optimization Problems in Complex Systems

TL;DR

This paper introduces the canonical duality-triality theory, providing a unified framework for solving complex nonconvex optimization problems and identifying global and local extrema, with proofs and practical examples.

Contribution

It proves the triality theory for sums of exponentials and quartic polynomials, solving an open problem from 2003, and bridges global optimization with nonconvex mechanics.

Findings

Proved the triality theory for specific nonconvex functions

Demonstrated the theory's effectiveness through detailed examples

Provided a method to find global minima and local extrema

Abstract

General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples.