Global optimal solutions to nonconvex optimisation problems with a sum of double-well and log-sum-exp functions

This paper presents a canonical dual approach for solving a nonconvex global optimisation problem with a sum of double-well and log-sum-exp functions. Such a problem arises extensively in mechanics, robot designing, information theory and network communication systems. It includes fourth-order polynomial minimisation problems and minimax problems. Based on the canonical duality theory, this nonconvex problem is transformed to an equivalent dual problem, and the triality theory explicates that under certain condition the dual problem can be solved easily and, correspondingly, the global solution of the primal problem can be obtained analytically from the dual solution. It also discusses the relationships between local extremums of the primal problem and the dual problem. Furthermore, two specific problems, a fourth-order polynomial minimisation problem and a minimax problem, are discussed and situations when the condition in the triality theory holds are presented. In the end, several numerical examples are provided to illustrate the application of canonical duality theory on this problem.