Applied Variational Analysis: Structure, Regularity, and Algorithms

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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.