Canonical duality theory and triality for solving general global optimization problems in complex systems

Abstract

General nonconvex optimization problems are studied by using the canonical duality-triality theory. The triality theory is proved for sums of exponentials and quartic polynomials, which solved an open problem left in 2003. This theory can be used to find the global minimum and local extrema, which bridges a gap between global optimization and nonconvex mechanics. Detailed applications are illustrated by several examples.

RETRACTED: Double well potential function and its optimization in the n-dimensional real space: part II

“…In particular, if F(t, χ ) is quadratic, then (X) is a double-well-type fourth-order polynomial function, and is considered to be NP-hard in global optimization even for d = 1 (one-dimensional systems) [114,124]. However, by simply using the quadratic geometrical operator ξ k = (χ k ) = F(t k , χ k ), the nonconvex least squares problem (87) can be solved by the canonical duality-triality theory to obtain global optimal solution. Applications have been given to the logistic map [125] and population growth problems [126].…”

RETRACTED: Canonical duality theory for solving nonconvex/discrete constrained global optimization problems

“…It was realized in 2003 that the double-min duality (32) holds under certain additional condition [9,84]. Recently, it has been proved that this additional condition is simply dimχ = dimξ * to have the strong canonical double-min duality (32), otherwise, this double-min duality holds weakly in subspaces of X o × S o [85][86][87][88].…”

“…All of these functions appear extensively in modeling real-world problems, such as computational biology [127], bio-mechanics, phase transitions [29], filter design [132], location/transportation and networks optimization [129,130], communication and information theory (see [133]), etc. By using the canonical duality-triality theory, these problems can be solved nicely (see [87,[134][135][136][137][138][139][140][141][142][143]). …”

“…As a corollary of an iso-index theorem given in Gao's book, the double-min and double-max duality statements were proved for nonconvex optimization problems in finite-dimensional space (see Theorem 5.3.6 in [8]). Recently, this double-min duality has been proved for polynomial optimization [85,87,88], and for general global optimization problems [86,134]. The "certain additional constraints" are simply the dimensions of the primal problem and its canonical dual should be the same in order to have strong double-min duality.…”

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

“…The Gao-Strang total complementary function X is indeed useful for global optimization problems in real-world applications, at least for the problem (P) studied in this work. The results presented in this paper can be used for solving general non-convex constrained problems as long as the non-convex function W(x) can be written in the canonical form W(x) = V(L(x)) for certain quadratic geometrical operators L(x), such as the general sum of exponentials and polynomials [21]. It is worth continuing to study this topic both theoretically and numerically in order to develop efficient algorithms for solving challenging problems with general non-convex constraints.…”