Solutions and optimality criteria for nonconvex constrained global optimization problems with connections between canonical and Lagrangian duality

Abstract This paper presents a canonical duality theory for solving a general nonconvex 1 quadratic minimization problem with nonconvex constraints. By using the canonical dual 2 transformation developed by the first author, the nonconvex primal problem can be con3 verted into a canonical dual problem with zero duality gap. A general analytical solution 4 form is obtained. Both global and local extrema of the nonconvex problem can be identified 5 by the triality theory associated with the canonical duality theory. Illustrative applications 6 to quadratic minimization with multiple quadratic constraints, box/integer constraints, and 7 general nonconvex polynomial constraints are discussed, along with insightful connections 8 to classical Lagrangian duality. Criteria for the existence and uniqueness of optimal solutions 9 are presented. Several numerical examples are provided. 10