Variational Analysis: Theory, Algorithms, and Applications

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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.