Canonical duality for solving general nonconvex constrained problems

Abstract

This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and constraints possess certain patterns necessary for modeling real systems, a perfect dual problem (without duality gap) can be obtained in a unified form with global optimality conditions provided. While the popular augmented Lagrangian method may produce more difficult nonconvex problems due to the nonlinearity of constraints.

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

“…Thus, from the canonical transformation, we have τ = 2(∇χ) · (∇ C V (C(∇χ))) = ∇ F U(∇χ) (75) due to the chain role. This shows that the integral (71) is indeed a stationary point of (χ ) since τ ∈ T c .…”

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

“…This generalized canonical duality plays an important role in unified understanding Lagrangian duality and KKT theory for constrained optimization problems (see [74,75] and Section 5.4). In analysis, nonlinear PDEs are classified as semilinear, quasi-linear, and fully nonlinear three categories based on the degree of the nonlinearity [76].…”

RETRACTED: Canonical dual approach for contact mechanics problems with friction

“…Canonical duality theory, developed from non-convex analysis and global optimization [21,22], is a potentially powerful methodology, which has been successfully used for solving a large class of challenging problems in biology, engineering, sciences [23][24][25], and recently in network communications [26][27][28], radial basis neural networks [29] and constrained optimization [30]. In this paper we use this canonical dual transformation methodology in order to formulate the total complementarity function of the original problem whose stationary points do not have any duality gap with respect to the corresponding solutions of the primal problem.…”