Variational Analysis: Theory and Applications

变分分析：理论与应用

• 批准号: 0072179
• 负责人: Boris Mordukhovich
• 金额: $ 15.5万
• 依托单位: Wayne State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2003-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072179&HistoricalAwards=false
• 关键词: Variational; Analysis; Theory; Applications

1. The proposed research mainly deals with developing the first-order generalized differential theory of variational analysis in broad classes of infinite-dimensional spaces, with some basic aspects of the second-order variational analysis in finite dimensions, and with their applications to important problems arising in optimization, control, mechanics, and economics. The main tools of the generalized differentiation in the proposed research are based on the normal cones, subdifferentials, and coderivatives of nonsmooth objects, in the study of which the PI has been involved for a long time. The project will pay a particular attention to the second-order subdifferential theory and its applications to the Lipschitzian stability of variational systems, mathematical programs with equilibrium constraints, and to some problems of continuum mechanics. It will contain new development and applications in the area of dynamic optimization for control systems governed by differential, delay-differential, and partial differential equations/inclusions. It also aims to develop new applications of variational analysis to the study of Pareto optimality in equilibrium models of welfare economics. 2. Variational analysis has been recognized as a fruitful area in the modern Applied Mathematics that, on one hand, is concerned with finding the best solutions in large-scale mathematical models with complicated ``nonsmooth" constraints and, on the other hand, develops optimization strategies and technologies to solve a broad spectrum of real-life problems arising in various areas of applied science, control, economics, engineering, mechanics, etc. The proposed research aims to develop new methods of variational analysis and its application to some important problems in optimal control, economics, and mechanics. In particular, the proposed methods of minimax control design are largely motivated by applications to environmental systems. The proposed study of variational stability for optimization problems with equilibrium constraints is motivated by applications to practical problems of material design in continuum mechanics. New applications of variational analysis will be developed to multiobjective models of welfare economics.

1.拟议的研究主要涉及发展一阶广义微分理论的变分分析在广泛的类的无限维空间，与一些基本方面的二阶变分分析在有限维，并与他们的应用中出现的重要问题，在优化，控制，力学和经济学。广义微分的主要工具，在拟议的研究是基于正常的锥，次微分，和非光滑对象的余导数，在研究中，PI已经参与了很长一段时间。 该项目将特别关注二阶次微分理论及其在变分系统的Lipschitz稳定性，具有平衡约束的数学程序以及连续介质力学的一些问题中的应用。它将包含微分，延迟微分和偏微分方程/夹杂物控制系统的动态优化领域的新发展和应用。它还旨在发展新的应用变分分析的研究帕累托最优的福利经济学的均衡模型。 2.变分分析是现代应用数学中的一个富有成果的领域，它一方面致力于在具有复杂“非光滑”约束的大规模数学模型中寻找最佳解，另一方面发展优化策略和技术，以解决应用科学、控制、经济、工程、力学、本研究旨在发展变分分析的新方法及其在最优控制、经济学和力学中的一些重要问题的应用。特别是，极大极小控制设计的建议方法在很大程度上是由环境系统的应用程序。提出的研究变分稳定性的平衡约束的优化问题的动机是在连续介质力学的材料设计的实际问题的应用。变分分析的新应用将发展到福利经济学的多目标模型。