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

david Y. Gao（弗吉尼亚理工大学）摘要：全局优化问题在现实世界系统的数学建模中广泛存在，具有非常广泛的应用。由于代价函数的非凸性，全局优化中的许多问题都是np困难的。传统的直接方法和局部优化方法不能保证全局最小值的识别。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提供潜在的重要影响。本课题的主要目标是发展一种通用的正则原对偶方法及其相关的全局优化三性理论。本课题主要研究工程力学中一大类非凸变分问题的数值离散化直接引起的一般约束全局优化问题。第二个目标是推动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（零对偶间隙）；(2)提出了一种有趣的三角性理论，它可以作为非凸函数全局极小值和局部极值的最优性准则。每一个方面都有自己的挑战。完美对偶问题的表述需要非标准的方法，因为传统的对偶理论在凸分析中可能导致当原问题为非凸时出现所谓的对偶间隙。可行空间上非凸/非光滑函数的全局最优性判据由于其凹凸性和光滑性的损失而需要特殊的数学技术。由于所提出的问题在全局优化、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出现，该项目的广泛影响可能非常大。经典对偶变换的一般方法论和美丽三性理论将弥补这些领域之间存在的差距。软件和出版物的广泛分发（包括b施普林格即将出版的一套三卷本的《工程科学对偶理论手册》）将使工程、数学、物理和计算科学领域的实践者和科学家受益。将分别于2005年和2006年组织两次关于全局优化和非凸力学的国际会议。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个丰富的研究领域。这项提议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系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.