Variational Analysis: Theory, Algorithms, and Applications

这个项目的目的是研究优化和控制的数学方面。研究将集中在控制系统、社会经济学、机器人学、纳米技术和行为科学的工程设计中的实用模型。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。了解移动机器人建模、纳米技术中纳米粒子的动力学模型以及海洋水面车辆建模对国防尤为重要。这项研究将为培养博士生提供机会，包括那些来自代表性不足群体的博士生。这项研究的目的是发展变分分析方法和广义微分，强调数值算法和应用。这项研究将集中在一阶和二阶变分技术的广泛的理论和算法问题上，并应用于各种问题。具体地说，研究者将集中在有限和无限维上具有光滑和非光滑数据的约束优化，双层规划和设施选址，变分不等式和平衡系统，动态优化和非规则约束最优控制中的新的挑战性问题，抛物型拟变分不等式，以及非光滑控制系统的稳定性和反馈镇定。一阶和二阶广义微分的结果将适用于设计新的数值算法来解决优化、控制和相关问题。这些算法将在实际模型中实施。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.