Primal-Dual Method and Algorithm for large Scale Computation with Applications in Engineering Mechancis

弗吉尼亚理工学院和州立大学大规模计算的原对偶方法和算法及其在工程机械中的应用。高（弗吉尼亚理工大学）摘要：全局优化问题广泛存在于真实的世界系统的数学建模中，具有非常广泛的应用。由于代价函数的非凸性，许多全局优化问题是NP难的。传统的直接法和局部优化方法不能保证全局最优解的确定。另一方面，对偶理论和方法可能对全局优化问题的解和算法产生潜在的重要影响。 典范原对偶方法及其相关 全局优化中的试探理论该项目的重点是 一般约束全局优化问题，它直接产生于工程力学中的一大类非凸变分问题的数值离散化。第二个目标是推进某些强大的原始-对偶算法的发展，并将其具体应用于非凸力学.所提出的工作的智力价值包括：（1）一个潜在的强大的正则对偶变换，可以用来制定完美的对偶问题（2）一个有趣的试探性理论，它可以作为判别非凸函数全局极小和局部极值的最优性准则。 每一个方面都提出了自己的挑战。完全对偶问题的公式化需要非标准的方法，因为当原始问题是非凸的时，凸分析中的传统对偶理论可能导致所谓的对偶间隙。可行空间上的非凸/非光滑函数的全局最优性准则由于其凸性和光滑性的损失而需要特殊的数学技巧，由于所提出的问题涉及全局最优化、非凸/非光滑分析、工程力学、现代材料、数学物理和科学计算等多学科领域，因此该项目的广泛影响是非常巨大的。正则对偶变换的一般方法论与美 试用理论将弥合这些领域之间现有的差距。软件和出版物的广泛分发（包括一套三卷的工程科学对偶理论手册将由施普林格出版）将使工程，数学，物理和计算科学的从业者和科学家受益。两次关于全局优化和非凸力学的国际会议 将分别于2005年和2006年举办。这些会议将开启现代分析、优化和工程科学的新趋势。此外，他们将激励年轻的教师和学生冒险进入这一丰富的研究领域。 这项拟议中的工作将由弗吉尼亚理工大学的一个跨学科团队进行，该团队包括来自数学和工程系的本科生和研究生。该项目的完成将为全局优化、非凸力学和计算科学奠定基础。

ABSTRACT0514768Virginia Polytechnic Institute and State UniversityPrimal-Dual Method And Algorithm For Large Scale Computation With Applications In Engineering MechancisDavid Y. Gao (Virginia Tech)Abstract: 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.