David Y. Gao
University of Ballarat, Australia
Title : Canonical Duality and Triality: Unified Modeling, Theory, and Algorithm for Solving Challenging Problems in Complex Systems
Abstract :

Duality is a beautiful, inspiring, and fundamental concept that underlies all natural phenomena. In mathematical economics, dynamical systems, nonlinear analysis, global optimization, control theory, management and decision science, industrial and systems engineering, numerical methods and scientific computation, duality principles and methods are playing more and more important roles. The 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 real-world applications. The associated triality theory reveals an interesting multi-scale duality pattern in complex systems, which can be used to identify both global and local extrema and to design powerful algorithms for solving challenging problems.

Beginning with multi-level dualities  in the Garden of Eden and I-Ching, the speaker will show a unified structure and splendid beauty in mathematics, physics, systems theory, fine art, religion, and linguistic, etc. By using a very simple minimization problem in quadratic programming, the speaker will present a mathematical theory of duality and its role for modeling complex phenomena and solving challenging problems in general nonconvex/nonsmooth/discrete systems. He will show that by this canonical duality theory, a class of nonlinear partial differential equations can be transformed to a unified algebraic (tensor) equation, which can be solved completely (for certain problems) to obtain all possible solutions, therefore, a unified analytical solution form can be obtained for a large class of problems in nonlinear analysis and global optimization; both global and local optimal solutions can be identified by a triality theory. Results will show that for many nonconvex variational problems, the global optimal solutions are usually nonsmooth, and cannot be captured by any traditional Newton-type direct approaches. Applications will be illustrated by challenging problems in engineering mechanics, operations research, as well as certain well-known NP-hard problems in computational sciences, including phase transitions, control of chaotic systems, general polynomial minimization, mixed integer and network optimization.

The speaker hopes this talk will bring some fundamentally new insights into complex systems theory.

Biography :

Professor David Y. Gao received his B.A. in Manufacturing/Material Science, M.A. in Aerospace Engineering. His PhD was obtained from Tsinghua University in Engineering Mechanics and Applied Math. Since then, he has held research and teaching positions in different institutes including MIT, Yale, Harvard, the University of Michigan, and Virginia Tech. Currently, he is the Alexander Rubinov Chair Professor at the University of Ballarat, and a Research Professor of Engineering Science at the Australian National University.

Professor Gao’s research interests range over theoretical and engineering mechanics,  nonlinear/nonconvex analysis, operations research, computational science, modeling, simulation, optimization and control of complex systems. He has published one research monograph (454 pp), one handbook, seven books, and about 150 scientific and philosophic papers. His main research contributions include a canonical duality-triality theory, several mathematical models in engineering mechanics and material science, a series of complete solutions to a class of nonconvex/nonsmooth/discrete problems in nonlinear sciences, and some deterministic methods/algorithms for solving certain NP-hard problems in global optimization and computational science. One application of this canonical duality theory in large deformation solid mechanics solved a 50-years open problem and leads to a pure complementary energy principle (i.e. the Gao Principle  in the literature), which has broad applications in engineering mechanics and physics.  One of the large deformed beam models he proposed in 1996 is now recognized as the nonlinear Gao beam which can be used to study post-buckling analysis and plays an important role in real-world applications. In discrete systems, this canonical duality theory shows that the NP-hard 0-1 integer programming problems are identical to a continuous unconstrained Lipschitzian global optimization problem which can be solved deterministically (see here).  

Professor Gao is a founding editor for Springer book series of Advances in Mechanics and Mathematics, and Taylor & Francis book series of Modern mechanics and Mathematics. He serves as an associate editor for several journals of applied math, optimization, solid mechanics, dynamical systems, and industrial and management engineering. Currently he is the Secretary-General and Vice President of International Society of Global Optimization http://isogop.org/

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