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Triality: God Image for Multi-Scale Complex Systems and Unified Theory for Identifying Chaos and NP-Hardness
April 3, 2017 @ 1:00 pm - 2:00 pm
Chaos in complex systems and NP-Hard problems in computer science have been studied extensively for more than fifty years, but most people may not know what the chaos really is, which problems are truly NPhard, and the inner connection of these two concepts. This lecture intends to answer these fundamental questions as well as to present a unified theory for modelling natural phenomena and a powerful methodology for solving challenging problems in multi-scale complex systems. Canonical duality is a breakthrough methodological theory, which can be used not only to model complicated phenomena within a unified framework, but also for solving a wide class of nonconvex/nonsmooth/discrete problems in multidisciplinary fields [1,2,3]. 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. Starting from dualities in Garden of Eden and traility in DNA and quantum mechanics, the speaker will first present a unified mathematical model for multi-scale complex systems, which lays a foundation for understanding complicated phenomena in nature, including bifurcation, chaos, decision making, game theory, information technology, logistics, manufactures, phase transitions, scheduling, and management science, etc. Based on this model, he will show how a precise mathematical theory of duality was developed and why this theory can be used for solving not only the most challenging problems in chaotic dynamics and post-buckling of nonlinear structures, but also a large class of so-called “NP-hard” problems in global optimization and computer science. The inner connection between this theory and other popular methodologies, such as SDP method in nonconvex/integer programming and HQ regularization in image process, will be discussed. Applications will be illustrated by some well-known benchmark problems in global optimization, sensor networks and bi-level topology optimization of structural design. A very interesting relation between chaos in nonlinear dynamics and NP-Hardness in global optimization will be revealed. This talk will bring some fundamental new insights into nonlinear sciences, global optimization, complex systems, and computational mathematics.
Professor David Y. Gao received his Ph.D. in Engineering Mechanics and Applied Math from Tsinghua University. He has held research and teaching positions in different institutes including MIT, Yale, Harvard, and Virginia Tech. He moved to Australia in 2010 for his current position as the Alexander Rubinov Chair Professor in School of Applied Sciences at the Federation University Australia. Professor Gao is the author of 14 monograph/handbook/special volumes and about 200 research papers (> 50% are single authored) on applied mathematics, theoretical and computational mechanics, global optimization and operations research etc. 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. Professor Gao’s multidisciplinary research has been supported continuously by different programs at US National Science Foundation (NSF) and US Air Force Office for Scientific Research (AFOSR) before he moved to Australia in 2010. He is one of a few researchers in the southern hemisphere who receive research grants every year directly from the AFOSR Washington Office. Recently, Professor Gao’s canonical dualitytriality theory has been identified by AFOSR as a breakthrough research and his team has win two prestigious international grant awards with total US$600,000 for 2016-2020. Professor Gao is an editor-in-chief for Springer book series Advances in Mechanics and Mathematics and Taylor & Francis book series Modern Mechanics and Mathematics. He is also an associate editor of about eight international journals. Since 2000, Professor Gao has delivered over 40 keynote/plenary/invited lectures at international conferences and more than 60 colloquium talks at different universities and institutions. As a chair and co-chair, he has organized successfully about 10 world congress/conferences. Currently, he is serving as the Secretary-General and Vice President of the International Society of Global Optimization. Detailed information can be found at his web page: http://sitevm1.ballarat.edu.au/dgao/
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