- CARMA COLLOQUIUM
- Speaker: Prof David Gao, Alexander Rubinov Professor of Mathematics, The University of Ballarat
- Title: Optimization and Control of Complex Systems: Canonical Duality Approach
- Location: Room V206, Mathematics Building (Callaghan Campus) The University of Newcastle
- Access Grid Venue: RMIT
- Time and Date: 4:00 pm, Thu, 17th Jun 2010
Nonconvex/nonsmooth phenomena appear naturally in many complex systems. In static systems and global optimization problems, the nonconvexity usually leads to multi-solutions in the related governing equations. Each of these solutions represents certain possible state of the system. How to identify the global and local stability and extremality of these critical solutions is a challenge task in nonconvex analysis and global optimization. The classical Lagrangian-type methods and the modern Fenchel-Moreau-Rockafellar duality theories usually produce the well-known duality gap. It turns out that many nonconvex problems in global optimization and computational science are considered to be NP-hard. In nonlinear dynamics, the so-called chaotic behavior is mainly due to nonconvexity of the objective functions. In nonlinear variational analysis and partial differential equations, the existence of nonsmooth solutions has been considered as an outstanding open problem.
In this talk, the speaker will present a potentially useful canonical duality theory for solving a class of optimization and control problems in complex systems. Starting from a very simple cubic nonlinear equation, the speaker will show that the optimal solutions for nonconvex systems are usually nonsmooth and cannot be captured by traditional local analysis and Newton-type methods. Based on the fundamental definitions of the objectivity and isotropy in continuum physics, the canonical duality theory is naturally developed, and can be used for solving a large class of nonconvex/nonsmooth/discrete problems in complex systems. The results illustrate the important fact that smooth analytic or numerical solutions of a nonlinear mixed boundary-value problem might not be minimizers of the associated variational problem. From a dual perspective, the convergence (or non-convergence) of the FDM is explained and numerical examples are provided. This talk should bring some new insights into nonconvex analysis, global optimization, and computational methods.
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