Partly Smooth Models and Algorithms
Optimization 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.
Lewis, Adrian S.
Davis, Damek Shea; Bindel, David S.
Operations Research and Information Engineering
Ph. D., Operations Research and Information Engineering
Doctor of Philosophy
Attribution 4.0 International
dissertation or thesis
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