Advances in Canonical Duality Theory and Applications in Global Optimization and NoncOnvex Systems(Abstract)

摘要

Duality is an inspiring and fundamental concept that underlies almost all natural phenomena. In mathematical science, dynamical systems, modern mechanics, global optimization, economics, control theory, numerical methods and scientific computation, duality principles and methods are playing more and more important roles. Duality theory and methods in convex systems have been well studied. However, in nonconvex systems, these well developed duality theory and methods usually lead to a so-called duality gap. Many non-convex/nonsmooth problems in global optimization are NP-hard. In nonconvex dynamical systems, traditional direct methods may produce the so-called chaotic solutions. Canonical duality theory is a newly developed, potentially powerful methodology, which is composed mainly of a canonical dual transformation and a triality theory. The canonical dual transformation can be used to formulate perfect dual problems without duality gap, while the triality theory reveals an interesting duality pattern in gen ral non-convex system and plays a fundamental role in nonlinear analysis and global optimization. In this talk, the speaker will present a review and some new developments on the canonical duality theory and its applications in global optimization and nonconvex analysis. It will show that by using the canonical dual transformation, many well-known non-convex/nonsmooth problems in high dimensional space can be reformulated into certain smooth canonical dual problems in lower dimensional space; integer programming problems can be converted to certain continuous dual problems; a large class of constrained nonlinear optimization problems can be assembled into a unified framework. Nonlinear differential equations are equivalent to certain algebraic systems. An insightful relation between the canonical dual transformation and nonlinear (or extended) Lagrange multiplier methods is presented. The triality theory can be used to identify both global and local optimizers, to control chaotic behavior of nonlinear systems, and to develop some potentially powerful algorithms for solving a large class of challenging problems. Extensive applications will be illustrated by general nonconvex constrained problems in global optimization and nonconvex analysis. Complete solutions to certain challenging problems will be presented including a class of nonconvex/nonsmooth variational/boundary value problems in mathematical physics. The speaker will also show a very interesting phenomenon in natural science, i.e. many different problems share the same canonical duality form. This talk should bring some fundamental insights into nonconvex systems and global optimization.