Methods of Variational Analysis in Optimization, Equilibria, and Control

优化、平衡和控制中的变分分析方法

• 批准号: 0603846
• 负责人: Boris Mordukhovich
• 金额: $ 30万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2010-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603846&HistoricalAwards=false
• 关键词: Methods; Variational; Analysis; Optimization; Equilibria

The project deals with developing new methods of variational analysis and their applications to various problems in optimization, equilibria, and control. Since modern variational principles and techniques intrinsically relate to models and problems with nonsmooth data, a large part of the research concerns generalized differentiation theory of the first and second order for nonsmooth functions, sets, and set-valued mappings. The obtained methods and results of generalized differentiation will be applied to deriving new optimality and suboptimality conditions and sensitivity/stability characterizations in problems of mathematical programming, optimization and equilibrium problems with equilibrium constraints, optimal control of evolution inclusions and partial differential and functional differential equations, etc. The primary goal of this project is to develop applications of advanced techniques of optimization, variational analysis, and optimal control to practical problems arising in mechanics, economics, engineering, environmental science, etc. This is a very challenging issue, which requires the development of new mathematical methods dealing with non-classical objects and models as well as with complex control systems. The research will particularly concern models of welfare economics with public environment and also economies involving oligopolistic markets, feedback design of control systems functioning under uncertainty, and other applications to real-life problems.

该项目涉及发展新的变分分析方法及其在优化，平衡和控制中的各种问题的应用。由于现代变分原理和技术本质上与非光滑数据的模型和问题有关，因此大部分研究涉及非光滑函数、集合和集值映射的一阶和二阶广义微分理论。所得的方法和结果将应用于数学规划、平衡约束下的优化与平衡问题、演化内含物的最优控制、偏微分与泛函微分方程等问题中新的最优性和次最优性条件及灵敏度/稳定性表征的推导。该项目的主要目标是将优化、变分分析和最优控制等先进技术应用于力学、经济学、工程学、环境科学等领域的实际问题。这是一个非常具有挑战性的问题，它需要发展新的数学方法来处理非经典对象和模型以及复杂的控制系统。研究将特别关注公共环境的福利经济学模型，以及涉及寡头垄断市场的经济，在不确定性下运行的控制系统的反馈设计，以及对现实生活问题的其他应用。