Global optimization problems are widespread in the mathematical modelling of real world systems for a very broad range of applications. Due to the nonconvexity of the cost functions, many problems in global optimization are NP-hard. Traditional direct methods and local optimization procedures can not guarantee the identification of the global minima. On the other hand, duality theory and methods may provide potentially important influence on the solutions and algorithms for global optimization problems. The primary goal of this project is to develop a general canonical primal-dual method and its associated triality theory in global optimization. The project focuses on a general constrained global optimization problem, which arises directly from numerical discretization of a large class of nonconvex variational problems in engineering mechanics. The secondary goal is to advance the development of certain powerful primal-dual algorithms with concrete applications to nonconvex mechanics.
The intellectual merit of the proposed work includes (1) a potentially powerful canonical dual transformation which can be used to formulate perfect dual problems (with zero duality gap); (2) an interesting triality theory which can serve as an optimality criterion to identify both global minima and local extrema of nonconvex function. Each of these aspects presents its own challenges. Formulation of perfect dual problems requires nonstandard methods, as the traditional duality theory in convex analysis may lead to a so-called duality gap when the primal problem is nonconvex. Global optimality criterion for nonconvex/nonsmooth functions over feasible spaces will need special mathematical techniques due to the loss of convexity and smoothness.
The broader impacts of this project are potentially very great as the proposed problem arises in multi-disciplinary fields of global optimization, nonconvex/nonsmooth analysis, engineering mechanics, modern materials, mathematical physics, and scientific computation. The general methodology of canonical dual transformation and the beautiful triality theory will bridge the existing gaps among these fields. The broad distribution of software and publications (including a set of three volumes of Handbook of Duality Theory in Engineering Science to be published by Springer) will benefit practitioners and scientists across engineering, mathematics, physics, and computational science. Two international conferences on global optimization and nonconvex mechanics will be organized in 2005 and 2006, respectively. These conferences will open new trends in modern analysis, optimization, and engineering science. Furthermore, they will stimulate young faculty and students to venture into this rich domain of research. The proposed work will be carried out by an interdisciplinary team at Virginia Tech which includes both undergraduate and graduate students from math and engineering departments. The completion of this project will lay a ground work in global optimization, nonconvex mechanics, and computational science.
