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

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. Zȃlinescu and his coworker are addressed. The paper is ended with some open problems and future works in global optimization and nonconvex mechanics.

On unconstrained optimization problems solved using the canonical duality and triality theories

“…In this section we analyze results obtained by DY Gao and his collaborators in papers dedicated to unconstrained optimization problems, related to "triality theorems". The main tool to identify the papers where this class of problems are considered was to look in the survey papers [4] (which practically includes [5]), [7] (which is almost the same as [6]), [15] (which is very similar to [8]), [13] (which is the same as [12]), as well as in the recent book [10]. Though, in order to understand the chronology of the development of this topic let us quote first the following texts from [13, p. 40] (see also [12,p.…”

“…NP30]) and [20, p. 136] (see also [19, p. 5]), respectively: Q1 -"the triality was proposed originally from post-buckling analysis [42] in "either-or" format since the double-max duality is always true but the double-min duality was proved only in one-dimensional nonconvex analysis [49]", 2 Q2 -"the triality theorem was formed by these three pairs of dualities and has been used extensively in nonconvex mechanics [10,17] and global optimization [3,21,34]. However, it was realized in 2003 [12,13] that if the dimensions of the primal problem and its canonical dual are different, the double-min duality (30) needs "certain additional conditions". For the sake of mathematical rigor, the double-min duality was not included in the triality theory and these additional constraints were left as an open problem (see Remark 1 in [12], also Theorem 3 and its Remark in a review article by Gao [13]).…”

“…However, it was realized in 2003 [12,13] that if the dimensions of the primal problem and its canonical dual are different, the double-min duality (30) needs "certain additional conditions". For the sake of mathematical rigor, the double-min duality was not included in the triality theory and these additional constraints were left as an open problem (see Remark 1 in [12], also Theorem 3 and its Remark in a review article by Gao [13]). By the facts that the double-max duality ( 29) is always true and the double-min duality plays a key role in real-life applications, it was still included in the triality theory in the either-or form in many applications for the purposes of perfection in esthesis and some other reasons in reality."…”

Analytic Solutions to 3-D Finite Deformation Problems Governed by St Venant–Kirchhoff Material

“…This triality theory reveals an important fact in nonconvex analysis, i.e. for a given statically admissible field τ ∈ T a , if the canonical dual equation (29) The pure complementary energy principle and triality theory play a fundamental role not only in nonconvex analysis, but also in computational science and global optimization (see [15,17,18,22]).…”

“…The pure complementary energy principle and triality theory play a fundamental role not only in nonconvex analysis, but also in computational science and global optimization (see [15,17,18,22]).…”