Applied Variational Analysis: Structure, Regularity, and Algorithms

应用变分分析：结构、规律和算法

• 批准号: 0504032
• 负责人: Adrian Lewis
• 金额: $ 27.54万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-09-01 至 2008-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504032&HistoricalAwards=false
• 关键词: Applied; Variational; Analysis; Structure; Regularity

Traditional calculus is a central scientific tool for understanding phenomena varying smoothly in time and space. Variational analysis extends the reach of calculus beyond smooth scenarios to optimization problems in a wide range of modern disciplines, including management science, control engineering, and economics. This project bridges the gulf between an existing, well-developed theory on the one hand, and much less-developed concrete applications on the other. Nonsmooth geometry is challenging: unlike in calculus, the shapes and structures of interest have corners and sharp ridges, frustrating traditional mathematical and computational analysis. This project will relate the difficulty of solving a nonsmooth optimization problem to a crucial theoretical notion of its "regularity". In the longer term, the project will pursue effective computational solution algorithms for nonsmooth optimization. A fertile interplay between innovative theory, high-impact engineering applications, and reliable solution methods is central to the project's success. Graduate students, international dissemination, and interdisciplinary collaborations with institutes of computing, control engineering, and industrial sciences, will all play important roles in the project's development.

传统微积分是理解在时间和空间上平稳变化的现象的中心科学工具。变分分析将微积分的范围从平稳情景扩展到包括管理科学、控制工程和经济学在内的广泛的现代学科中的优化问题。这个项目在现有的、发展良好的理论和不太发达的具体应用之间架起了一座桥梁。非光滑的几何学具有挑战性：与微积分不同，感兴趣的形状和结构有角和尖锐的脊线，这阻碍了传统的数学和计算分析。这个项目将把解决非光滑最优化问题的难度与它的“正则性”这一关键的理论概念联系起来。从长远来看，该项目将寻求非光滑优化的有效计算求解算法。创新理论、高影响力的工程应用和可靠的解决方案方法之间的相互作用是该项目成功的关键。研究生、国际传播，以及与计算机、控制工程和工业科学研究所的跨学科合作，都将在该项目的发展中发挥重要作用。