Variational Analysis: Theory, Algorithms, and Applications

变分分析：理论、算法和应用

• 批准号: 2204519
• 负责人: Boris Mordukhovich
• 金额: $ 25万
• 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: 2022
• 资助国家: 美国
• 起止时间: 2022-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204519&HistoricalAwards=false
• 关键词: Variational; Analysis; Theory; Algorithms; Applications

This project aims to investigate mathematical aspects of optimization and control. The research will focus on practical models in engineering design of control systems, socioeconomics, robotics, nanotechnology, and behavioral sciences. Investigating and solving such models using advanced mathematical methods will promote the progress of science and enhance national security. Understanding mobile robot modeling, dynamic models for nanoparticles in nanotechnology, and marine surface vehicle modeling will be particularly important for national defense. The research will provide opportunities for training Ph.D. students, including those from underrepresented groups. The research aims to develop variational analysis methods and generalized differentiation, emphasizing numerical algorithms and applications. The research will focus on a broad spectrum of theoretical and algorithmic issues in first-order and second-order variational techniques with applications to various problems. Specifically, the investigator will focus on constrained optimization with smooth and nonsmooth data in finite and infinite dimensions, bilevel programming and facility location, variational inequalities and equilibrium systems, new challenging classes of problems in dynamic optimization and optimal control with irregular constraints, parabolic quasi-variational inequalities, and stability of nonsmooth control systems and feedback stabilization. The first-order and second-order generalized differentiation results will be applicable to design novel numerical algorithms to solve optimization, control, and related problems. The algorithms will be implemented in practical modeling.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.

本项目旨在研究优化和控制的数学方面。该研究将侧重于控制系统，社会经济学，机器人技术，纳米技术和行为科学的工程设计中的实用模型。利用先进的数学方法研究和求解这类模型，将促进科学进步，增强国家安全。了解移动的机器人建模、纳米技术中纳米颗粒的动态模型以及海洋表面车辆建模对国防将特别重要。该研究将为培养博士生提供机会。学生，包括来自代表性不足群体的学生。本研究旨在发展变分分析方法与广义微分，著重数值演算法与应用。该研究将集中在一阶和二阶变分技术与应用到各种问题的理论和算法问题的广泛范围。具体而言，调查员将专注于约束优化与光滑和非光滑数据在有限和无限维，双层规划和设施位置，变分不等式和平衡系统，新的挑战性的问题类动态优化和最优控制与不规则约束，抛物准变分不等式，和非光滑控制系统的稳定性和反馈稳定。一阶和二阶广义微分的结果将适用于设计新的数值算法，以解决优化，控制和相关问题。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

项目成果

A Generalized Newton Method for Subgradient Systems

• DOI: 10.1287/moor.2022.1320
• 发表时间: 2020-09
• 期刊: Mathematics of Operations Research
• 影响因子: 1.7
• 作者: Pham Duy Khanh;B. Mordukhovich;Vo Thanh Phat

A globally convergent proximal Newton-type method in nonsmooth convex optimization

• DOI: 10.1007/s10107-022-01797-5
• 发表时间: 2020-11
• 期刊: Mathematical Programming
• 影响因子: 2.7
• 作者: B. Mordukhovich;Xiaoming Yuan;Shangzhi Zeng;Jin Zhang

Globally convergent coderivative-based generalized Newton methods in nonsmooth optimization

• DOI: 10.1007/s10107-023-01980-2
• 发表时间: 2021-09
• 期刊: Math. Program.
• 作者: P. D. Khanh;B. Mordukhovich;Vo Thanh Phat;D. Tran

Generalized damped Newton algorithms in nonsmooth optimization via second-order subdifferentials

基于二阶亚微分的非光滑优化中的广义阻尼牛顿算法

• DOI: 10.1007/s10898-022-01248-7
• 发表时间: 2023
• 期刊: Journal of Global Optimization
• 影响因子: 1.8
• 作者: Khanh, Pham Duy;Mordukhovich, Boris S.;Phat, Vo Thanh;Tran, Dat Ba
• 通讯作者: Tran, Dat Ba

Variational Convexity of Functions and Variational Sufficiency in Optimization

• DOI: 10.1137/22m1519250
• 发表时间: 2022-08
• 期刊: SIAM J. Optim.
• 作者: P. D. Khanh;B. Mordukhovich;Vo Thanh Phat