Title:	Duality and Triality: Unifying Mathematical Modeling, Methods, and Computational Science
Date:	June 5, 2007 (Tuesday)
Time:	4:00 p.m. - 5:00 p.m.
Venue:	Room 121, 1/F, Ho Sin-hang Engineering Building, The Chinese University of Hong Kong, Shatin, N.T.
Speaker:	Professor David Yang Gao Virginia Tech

ABSTRACT:

Duality is an inspiring, fundamental concept that underlies almost all natural phenomena. In mathematical physics, engineering science, system and information theory, global optimization, economics, control theory, numerical methods and scientific computation, duality principles and methods are playing more and more important roles. Triality is a newly developed concept, which reveals an intrinsic duality pattern in general systems.

Beginning with dualities in the Garden of Eden, motivated by some interesting problems in physics and optimization, the speaker will reveal a unified canonical (one-to-one) duality structure and splendid beauty in mathematical physics, complex systems, aesthetics, and linguistic, etc. By using a very simple, but fundamentally difficult quadratic minimization problem, the speaker will present a potentially powerful canonical dual transformation method and the associated triality theory. He will first show that by using this method, some well-known nonconvex/nonsmooth variational problems can be converted into certain dual algebraic systems in finite dimensional space; a class of NP-hard problems in global optimization and computational science can be reformulated into certain simple canonical dual problems. Therefore, complete solutions can be obtained. For those difficult problems in chaotic dynamical systems and phase transitions, the speaker will explain the reason that leads to chaos, and why the traditional direct methods can not be used along to solve nonconvex problems. He will show that by using the canonical dual transformation, chaotic trajectories in phase space form an invariant set in dual phase space. The triality theory can be used to control the chaotic behavior of the nonconvex systems, and to identify both global minimizer and local extrema. Based on this triality theory, powerful primal-dual algorithms can be developed for solving large-scale, multi-scale nonlinear problems.

Applications will be illustrated by solving some well-known problems in nonconvex analysis, global optimization, and mathematical physics. Complete solutions to certain well-known difficult global optimization problems will be presented, including polynomial minimization, integer programming, Boolean least square problems. Numerical simulations for a class of nonconvex variational problems in phase transition, chaotic dynamics will be demonstrated by movie. This talk should bring some new insights into nonconvex systems and computational science.

BIOGRAPHY:

Professor David Gao received his Ph.D. from Tsinghua University and did post-doctoral research at MIT, Yale, and Harvard University before joining the Virginia Tech. His research interests range over applied mathematics, engineering mechanics, global optimization, and computational science. His main research contribution is the canonical duality theory in general nonconvex systems. He has published seven books and about 90 papers. Currently he serves as an Editor-in-Chief of the Handbook of Duality Principles (Springer), a co-Editor-in-Chief of two book series (Springer and Taylor & Francis), as well as an associate editor for about five international journals.

Enquiries: Miss Temmy So at tel 2609 8444

For more information, please refer to http://www.cse.cuhk.edu.hk/seminar