Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex Systems

TL;DR

This paper reviews the canonical duality-triality theory, a unifying framework that bridges nonconvex analysis, mechanics, and global optimization, addressing complex systems and challenging real-world problems.

Contribution

It provides a comprehensive overview of the theory's foundations, its role in connecting nonconvex analysis with optimization, and addresses recent misunderstandings and open problems.

Findings

Bridges gap between nonconvex mechanics and global optimization

Addresses NP-hard problems with new theorems and algorithms

Clarifies misconceptions about key concepts in the field

Abstract

Canonical duality-triality is a breakthrough methodological theory, which can be used not only for modeling complex systems within a unified framework, but also for solving a wide class of challenging problems from real-world applications. This paper presents a brief review on this theory, its philosophical origin, physics foundation, and mathematical statements in both finite and infinite dimensional spaces, with emphasizing on its role for bridging the gap between nonconvex analysis/mechanics and global optimization. Special attentions are paid on unified understanding the fundamental difficulties in large deformation mechanics, bifurcation/chaos in nonlinear science, and the NP-hard problems in global optimization, as well as the theorems, methods, and algorithms for solving these challenging problems. Misunderstandings and confusions on some basic concepts, such as objectivity, nonlinearity, Lagrangian, and generalized convexities are discussed and classified. Breakthrough from recent challenges and conceptual mistakes by M. Voisei, C. Zalinescu and his co-worker are addressed. Some open problems and future works in global optimization and nonconvex mechanics are proposed.