Canonical Duality Theory: Connections between nonconvex mechanics and global optimization

This paper presents a comprehensive review and some new developments on canonical duality theory for nonconvex systems. Based on a tri-canonical form for quadratic minimization problems, an insightful relation between canonical dual transformations and nonlinear (or extended) Lagrange multiplier methods is presented. Connections between complementary variational principles in nonconvex mechanics and Lagrange duality in global optimization are also revealed within the framework of the canonical duality theory. Based on this framework, traditional saddle Lagrange duality and the so-called bi-duality theory, discovered in convex Hamiltonian systems and d.c. programming, are presented in a unified way; together, they serve as a foundation for the triality theory in nonconvex systems. Applications are illustrated by a class of nonconvex problems in continuum mechanics and global optimization. It is shown that by the use of the canonical dual transformation, these nonconvex constrained primal problems can be converted into certain simple canonical dual problems, which can be solved to obtain all extremal points. Optimality conditions (both local and global) for these extrema can be identified by the triality theory. Some new results on general nonconvex programming with nonlinear constraints are also presented as applications of this canonical duality theory. This review brings some fundamentally new insights into nonconvex mechanics, global optimization, and computational science.