Variational Analysis of Optimal Value Functions and Applications to Nonsmooth Optimization

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Variational analysis serves as the mathematical foundation for non-smooth optimization problems in which the cost functions to be minimized are not necessarily differentiable. Because of the non-differentiability, traditional calculus-based methods are not applicable. Through this research project, the principal investigator and his colleagues will develop new applications of variational analysis designed to solve a number of important non-smooth optimization problems in the areas of facility location, computational geometry, and machine learning. They will develop and implement numerical algorithms for large-scale location problems, some involving different types of distance metrics, etc. The methods being built will be used to study other non-smooth optimization models in computational geometry and machine learning. The new knowledge in variational analysis this project anticipates will advance the solution of practical models in non-smooth optimization.This project aims at developing new applications of variational analysis to non-smooth optimization. The principal investigator and his colleagues study generalized differentiation properties of a class of optimal value functions in both convex and non-convex settings. Functions of this type, are intrinsically non-differentiable, and play an important role in the theory of variational analysis and its applications. In particular, the PI and his colleagues focus on two classes of optimal value functions: the minimal time function, which is a natural extension of the closest distance function, and the maximal time function, which is an extension of the farthest distance function. Generalized differentiation properties of the optimal value function are used to study necessary and sufficient conditions on initial data that guarantee different properties of the optimal value function such as continuity, Lipschitz continuity, and differentiability. Results obtained here contribute to development of numerical algorithms for the solution of non-smooth optimization problems in facility location, computational geometry, and machine learning. Generalized differentiation properties of the optimal value function as well as advanced smoothing techniques and fast gradient methods are investigated in order to develop effective numerical algorithms for solving these problems.