Canonical duality theory: Unified understanding and generalized solution for global optimization problems

Canonical duality theory is a potentially powerful methodology, which can be used to model complex systems with a unified solution to a wide class of discrete and continuous problems in global optimization and nonconvex analysis. This paper presents a brief review and recent developments of this theory with applications to some well-know problems, including polynomial minimization, mixed integer and fractional programming, nonconvex minimization with nonconvex quadratic constraints, etc. Results shown that under certain conditions, these difficult problems can be solved by deterministic methods within polynomial times, and NP-hard discrete optimization problems can be transformed to certain minimal stationary problems in continuous space. Concluding remarks and open problems are presented in the end.