Applied Variational Analysis: Structure, Regularity, and Algorithms

传统微积分是理解时空平滑变化现象的核心科学工具。变分分析将微积分的范围从平滑的场景扩展到广泛的现代学科中的优化问题，包括管理科学、控制工程和经济学。这个项目一方面弥合了现有的、发展良好的理论与另一方面发展较差的具体应用之间的鸿沟。非光滑几何是具有挑战性的：与微积分不同，感兴趣的形状和结构有角和尖锐的脊，使传统的数学和计算分析受挫。这个项目将把解决一个非光滑优化问题的困难与它的“规律性”的一个关键理论概念联系起来。从长远来看，该项目将追求有效的非光滑优化计算解决算法。创新理论、高影响力工程应用和可靠的解决方案方法之间的丰富相互作用是项目成功的关键。研究生、国际传播以及与计算机、控制工程和工业科学研究所的跨学科合作，都将在项目的发展中发挥重要作用。

Traditional calculus is a central scientific tool for understanding phenomena varying smoothly in time and space. Variational analysis extends the reach of calculus beyond smooth scenarios to optimization problems in a wide range of modern disciplines, including management science, control engineering, and economics. This project bridges the gulf between an existing, well-developed theory on the one hand, and much less-developed concrete applications on the other. Nonsmooth geometry is challenging: unlike in calculus, the shapes and structures of interest have corners and sharp ridges, frustrating traditional mathematical and computational analysis. This project will relate the difficulty of solving a nonsmooth optimization problem to a crucial theoretical notion of its "regularity". In the longer term, the project will pursue effective computational solution algorithms for nonsmooth optimization. A fertile interplay between innovative theory, high-impact engineering applications, and reliable solution methods is central to the project's success. Graduate students, international dissemination, and interdisciplinary collaborations with institutes of computing, control engineering, and industrial sciences, will all play important roles in the project's development.