Wylie, Calvin2020-06-232020-06-232019-12Wylie_cornellgrad_0058F_11824http://dissertations.umi.com/cornellgrad:11824https://hdl.handle.net/1813/70089126 pagesOptimization and variational problems typically involve a highly structured blend of smooth and nonsmooth geometry. In nonlinear programming, such structure underlies the design of active-set algorithms, in which a globally convergent process first simplifies the problem by identifying active constraints at the solution; a second phase then employs a rapidly-convergent Newton-type method, with linear models of the simplified problem playing a central role. The theory of partial smoothness formalizes and highlights the fundamental geometry driving ``identification.'' This dissertation concentrates on the second phase, and understanding accelerated local convergence in partly smooth settings. A key contribution is a simple algorithm for ``black-box'' nonsmooth optimization, that incorporates second-order information with the usual linear approximation oracle. Motivated by active sets and sequential quadratic programming, a model-based approach is used to prove local quadratic convergence for a broad class of objectives. Promising numerical results on more general functions, as well as simple first-order analogues, are discussed. Beyond optimization, an intuitive linearization scheme for generalized equations is formalized, with simple techniques based on classical differential geometry: manifolds, normal and tangent spaces, and constant rank maps. The approach illuminates fundamental geometric ideas behind active-set acceleration techniques for variational inequalities, as well as second-order theory and algorithms for structured nonsmooth optimization.enAttribution 4.0 Internationaldissertation or thesishttps://doi.org/10.7298/c7gg-3e23