Advanced Research on Second-Order Variational Analysis with New Applications to Optimization, Control, and Practical Modeling

Wayne State University, Detroit MI

Investigators

Abstract

This project is devoted to developing advanced tools of mathematical analysis and optimization of large-scale systems arising in various applications. Such systems include nonstandard optimization-related and equilibrium problems with data that may not be differentiable in the usual sense. Besides developing new mathematical knowledge, the principal investigator and his seven PhD students who participate in the project (including four students from underrepresented groups in the mathematical sciences) pay particular attention to a variety of models arising in applications from socioeconomics, traffic equilibria, behavioral science, and water resources. The project studies new topics in second-order variational analysis, optimization, and systems control that are largely motivated by problems arising in applications. It is conditionally divided into five interrelated parts. Part I concerns open problems in second-order generalized differential theory and its applications to optimization-related areas of nonlinear analysis. Particular attention is paid to constructive computations of major second-order generalized derivatives for remarkable classes of extended-real-valued functions which play a crucial role in the subsequent parts of the project. Part II is devoted to second-order characterizations of various stability notions, including tilt and full stability in nonpolyhedral conic programming, and elliptic variational inequalities. Part III of the project deals with the study of critical multipliers in variational systems that are largely responsible for the slow convergence of primal-dual algorithms of optimization. Here, the PI and his collaborators develop new Newton-type methods to solve nonsmooth optimization and related problems. Part IV is devoted to novel developments in control theory, including feedback control and stabilization of ODE and PDE systems. Another major topic considered is optimal control of nonconvex versions of the sweeping process that plays a key role in subsequent applications. Some of these applications are the focus of Part V of the project, where the PI and his collaborators undertake a comprehensive study of the optimization problems for the controlled planar crowd motion model, which is well recognized in socioeconomics and traffic equilibria. Other types of applications considered stem from behavioral sciences and water resource models. 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.