RETRACTED: Canonical duality–triality theory: bridge between nonconvex analysis/mechanics and global optimization in complex systems

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 co-worker are addressed. Some open problems and future works in global optimization and nonconvex mechanics are proposed.

RETRACTED: On the minimal distance between two non-convex surfaces

“…If l = 0, then from equation (22), y = z and so (y, z) 6 2 X c because Y c \ Z c = ;. Dually, if m6 ¼ 0 then from equation (25), L(z) = DV * ( §) which combined together with equations (19) and (24) provides z 2 Z c . Since y 2 Y c , from equation (23), it has been proven that x 2 X c .…”

On modeling and global solutions for d.c. optimization problems by canonical duality theory

“…Clearly, this generalized d.c. minimization problem can be used to model a reasonably large class of real-world problems in mathematical physics [9,11], global optimization [16], and computational sciences [19]. By the fact that V (ξ) is convex, l.s.c.…”

“…This theory was developed originally from Gao and Strang's work in nonconvex mechanics [21] and has been applied successfully for solving a large class of challenging problems in both nonconvex analysis/mechancis and global optimization, such as phase transitions in solids [23], post-buckling of large deformed beam [38], nonconvex polynomial minimization problems with box and integer constraints [13,15,18], Boolean and multiple integer programming [6,40], fractional programming [7], mixed integer programming [20], polynomial optimization [14], high-order polynomial with log-sum-exp problem [3]. A comprehensive review on this theory and breakthrough from recent challenges are given in [19]. The goal of this paper is to apply the canonical duality theory for solving the challenging d.c. programming problem (1).…”

“…The nonlinear transformation (19) is actually the canonical transformation, first introduced by Gao in 2000 [10], and ξ = Λ(x) is called a canonical measure. The canonical measure ξ = Λ(x) is also called the geometrically admissible measure in the canonical duality theory [10], which is not necessarily to be objective.…”

Analytic Solutions to Optimal Control Problems with Constraints

“…The canonical duality theory was developed from nonconvex analysis and mechanics during the last decade (see [9] [10]), and has shown its potential for global optimization and nonconvex nonsmooth analysis [10]- [14]. Meanwhile, we introduce a differential flow for constructing the so-called canonical dual function to deduce some optimality conditions for solving global optimizations, which is shown to extend some corresponding results in canonical duality theory [9]- [11]. In comparison to the previous work mainly focused on simple constraints, we not only discuss linear box constraints, but also the nonlinear sphere constraint.…”