Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex Systems
arXiv:1410.2665
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.
arXiv comment43 pages, 4 figures. appears in Mathematics and Mechanics of Solids, 2014
References in corpus (7)
- Cosmic Strings and Superstrings
- A novel canonical dual computational approach for prion AGAAAAGA amyloid fibril molecular modeling
- On the Triality Theory in Global Optimization
- Global solutions to a class of CEC benchmark constrained optimization problems
- A Potential Reduction Method for Canonical Duality, with an Application to the Sensor Network Localization Problem
- Canonical duality for solving general nonconvex constrained problems
- Analytical Solutions to General Anti-Plane Shear Problems In Finite Elasticity