Perfect duality theory and complete solutions to a class of global optimization problems*

Abstract

This article presents a complete set of solutions for a class of global optimization problems. These problems are directly related to numericalization of a large class of semilinear nonconvex partial differential equations in nonconvex mechanics including phase transitions, chaotic dynamics, nonlinear field theory, and superconductivity. The method used is the so-called canonical dual transformation developed recently. It is shown that, by this method, these difficult nonconvex constrained primal problems in can be converted into a one-dimensional canonical dual problem, i.e. the perfect dual formulation with zero duality gap and without any perturbation. This dual criticality condition leads to an algebraic equation which can be solved completely. Therefore, a complete set of solutions to the primal problems is obtained. The extremality of these solutions are controlled by the triality theory discovered recently [D.Y. Gao (2000). Duality Principles in Nonconvex Systems: Theory, Methods and Applications, Vol. xviii, p. 454. Kluwer Academic Publishers, Dordrecht/Boston/London.]. Several examples are illustrated including the nonconvex constrained quadratic programming. Results show that these problems can be solved completely to obtain all KKT points and global minimizers.