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2017 | OriginalPaper | Buchkapitel

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

verfasst von : David Yang Gao, Ning Ruan, Vittorio Latorre

Erschienen in: Canonical Duality Theory

Verlag: Springer International Publishing

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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. Particular emphasis is placed 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 confusion 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. Zălinescu and his coworker are addressed. The paper is ended with some open problems and future works in global optimization and nonconvex mechanics.

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Nächstes Kapitel Analytic Solutions to Large Deformation Problems Governed by Generalized Neo-Hookean Model
Fußnoten
1
Also known as the Book of Changes, Zhouyi and Yijing, is the world oldest and most sophisticated system of wisdom divination, the fundamental source of most of the eastern philosophy, medicine, and spirituality. Traditionally, it was believed that the principles of the I Ching originated with the mythical King Fu Xi during the 3rd and 2nd millennia BCE.
 
2
This terminology is used mainly in English literature. The function f(x) is called the target function in Chinese and Japanese literatures, the goal function in Russian and German literatures.
 
3
Tensor is a geometrical object which is defined as a multidimensional array satisfying a transformation law (see [120]). A tensor must be independent of a particular choice of coordinate system (frame-indifference). But this terminology has been misused in optimization literature, where, any multidimensional array of data is called tensor (see [6]).
 
4
It is an unfortunate truth that many people do not know the relation between the Lagrangian space \({\mathbb R}^n\) they work in and the Minkowski (physical) space \({\mathbb R}^3\times {\mathbb R}\) they live in.
 
5
The neighborhood \({\mathscr {X}}_o\) of \(\bar{{\varvec{\chi }}}\) means that on which, \(\bar{{\varvec{\chi }}}\) is the only stationary point.
 
7
The quasiconvexity used in variational calculus and continuum physics has an entirely different meaning from that used in optimization, where a function \(f:{\mathbb R}^n \rightarrow {\mathbb R}\) is called quasiconvex if its level set \({\mathscr {L}}_{{\alpha }}[f] = \{ x \in {\mathbb R}^n | \; f(x) \le {\alpha }\} \) is convex. For example, the nonconvex function \(f(x) = \sqrt{|x|}\) is quasiconvex.
 
8
The second Piola–Kirchhoff stress tensor is defined by \(\mathbf{T}= \partial \varPhi (\mathbf{E})\), where \(\mathbf{E}= \frac{1}{2}(\mathbf{C}- \mathbf{I})\) is the Green–St. Venant strain tensor. Therefore, we have \(\mathbf{S}= 2 \mathbf{T}\).
 
9
Clearly, we can adopt high-order rule for approximation of \(F(t,{Y})\) at the \(k-1\) step, which will should be subjected to study in the future.
 
10
Indeed, one authors’ paper [127] was first submitted to a computational optimization journal and received such a reviewer’s comment: “the authors applied a perturbation, which changed the problem mathematically, ... and I suggest an immediate rejection.”
 
11
This sentence is deleted by Voisei and Zălinescu in their revision of [137] after they were informed by referees that their counterexamples are not new and the triality theory has been proved.
 
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Metadaten
Titel
Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex System
verfasst von
David Yang Gao
Ning Ruan
Vittorio Latorre
Copyright-Jahr
2017
Buch
Canonical Duality Theory

Print ISBN: 978-3-319-58016-6

Electronic ISBN: 978-3-319-58017-3

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-58017-3_1

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