Advanced Research on Second-Order Variational Analysis with New Applications to Optimization, Control, and Practical Modeling

二阶变分分析的高级研究及其在优化、控制和实际建模中的新应用

基本信息

批准号：
1808978
负责人：
Boris Mordukhovich
金额：
$ 24万
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：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2023-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1808978&HistoricalAwards=false
关键词：
Advanced Research Second Order Variational

项目摘要

This project is devoted to developing advanced tools of mathematical analysis and optimization of large-scale systems arising in various applications. Such systems include nonstandard optimization-related and equilibrium problems with data that may not be differentiable in the usual sense. Besides developing new mathematical knowledge, the principal investigator and his seven PhD students who participate in the project (including four students from underrepresented groups in the mathematical sciences) pay particular attention to a variety of models arising in applications from socioeconomics, traffic equilibria, behavioral science, and water resources.The project studies new topics in second-order variational analysis, optimization, and systems control that are largely motivated by problems arising in applications. It is conditionally divided into five interrelated parts. Part I concerns open problems in second-order generalized differential theory and its applications to optimization-related areas of nonlinear analysis. Particular attention is paid to constructive computations of major second-order generalized derivatives for remarkable classes of extended-real-valued functions which play a crucial role in the subsequent parts of the project. Part II is devoted to second-order characterizations of various stability notions, including tilt and full stability in nonpolyhedral conic programming, and elliptic variational inequalities. Part III of the project deals with the study of critical multipliers in variational systems that are largely responsible for the slow convergence of primal-dual algorithms of optimization. Here, the PI and his collaborators develop new Newton-type methods to solve nonsmooth optimization and related problems. Part IV is devoted to novel developments in control theory, including feedback control and stabilization of ODE and PDE systems. Another major topic considered is optimal control of nonconvex versions of the sweeping process that plays a key role in subsequent applications. Some of these applications are the focus of Part V of the project, where the PI and his collaborators undertake a comprehensive study of the optimization problems for the controlled planar crowd motion model, which is well recognized in socioeconomics and traffic equilibria. Other types of applications considered stem from behavioral sciences and water resource models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目致力于开发数学分析和优化大规模系统的高级工具。这样的系统包括与数据相关的非标准优化相关和均衡问题，这些问题在通常的意义上可能无法差异。除了发展新的数学知识外，还参与该项目的主要研究者及其七个博士学位学生（包括数学科学中代表性不足的群体的四名学生）特别关注各种模型，这些模型在社会经济学，交通平衡，行为科学和水资源的应用中引起的应用。项目研究在第二阶变量分析中的新主题是二阶分析的新主题。它有条件地分为五个相互关联的部分。第一部分涉及二阶广义差异理论中的开放问题及其在非线性分析的优化领域的应用。特别关注主要的二级广义衍生物的建设性计算，用于在项目的后续部分中起着至关重要的作用。第二部分致力于各种稳定概念的二阶表征，包括非多层圆锥编程中的倾斜度和完全稳定性，以及椭圆形的变异不平等。该项目的第三部分介绍了在很大程度上导致原始二重算法优化算法缓慢收敛的变异系统中关键乘数的研究。在这里，PI和他的合作者开发了新的牛顿型方法来解决非平滑优化和相关问题。第四部分致力于控制理论的新发展，包括反馈控制和ODE和PDE系统的稳定。考虑的另一个主要主题是对随后的应用程序中关键作用的全面范围版本的最佳控制。其中一些应用是该项目V部分的重点，PI和他的合作者对受控平面人群运动模型的优化问题进行了全面研究，该模型在社会经济和交通平衡中得到了广泛认可。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的审查标准来评估的，因此被认为源于行为科学和水资源模型的其他类型的应用程序。