Second-Order Variational Analysis and Its Applications

该项目旨在开发有效的数学分析工具，以研究在变分分析的名义下统一的优化，系统控制和其他应用数学领域中出现的挑战性问题。在标准微积分方法不可用的情况下，这种新的分析形式采用了广义微分学。开发这些分析工具的主要动机是为了解决工程、力学和行为科学中的一些实际问题。该项目的成果为研究人员和实践者提供了新的机制，使解决问题成为可能，为实际问题提供有价值的见解。本课题的研究重点是二阶变分分析，无论从理论研究还是实际应用的角度来看，二阶变分分析都还没有得到充分的研究。它系统地发展了二阶分析的广义微分工具，并将其应用于非光滑优化及相关领域的数值算法的设计和论证。在其他研究方向中，该项目将二阶变分分析应用于广泛的有限维和无限维变分系统，这远远超出了优化和最优控制。结果将有利于应用数学和应用科学界。

This project aims to develop efficient tools of mathematical analysis to investigate challenging problems arising in optimization, systems control, and other fields of applied mathematics unified under the name of variational analysis. This new form of analysis employs generalized differential calculus when the standard calculus methods are not available. The main motivation for developing these tools of analysis is to solve a number of practical problems of engineering, mechanics, and behavioral sciences. Results of this project arm researchers and practitioners with new machinery that make it possible to solve problems that provide valuable insight to practical issues.This project focuses on second-order variational analysis, which has been largely underinvestigated from both viewpoints of theoretical studies and practical applications. It carries out a systematic development of generalized differential tools of second-order analysis, with applications to the design and justification of numerical algorithms in nonsmooth optimization and related areas. Among other directions of research, the project applies second-order variational analysis to broad classes of finite-dimensional and infinite-dimensional variational systems, which go far beyond optimization and optimal control. Results will be beneficial to both applied mathematics and applied science communities.