Nonsmooth optimization consists of minimizing a continuous function by systematically choosing iterative points from the feasible set via the computation of function values and generalized gradients (called subgradients). Broadly speaking, this thesis contains two research themes: nonsmooth optimization algorithms and theories about the substructure of special nonsmooth functions. Specifically, in terms of algorithms, we develop new bundle methods and bundle trust region methods for generic nonsmooth optimization. For theoretical work, we generalize the notion of U-Lagrangian and investigate its connections with some subsmooth structures. This PhD project develops trust region methods for generic nonsmooth optimization. It assumes the functions are Lipschitz continuous and the optimization problem is not necessarily convex. Currently the project also assumes the objective function is prox-regular but no structural information is given. Trust region methods create a local model of the problem in a neighborhood of the iteration point (called the `Trust Region'). They minimize the model over the Trust Region and consider the minimizer as a trial point for next iteration. If the model is an appropriate approximation of the objective function then the trial point is expected to generate function reduction. The model problem is usually easy to solve. Therefore by comparing the reduction of the model's value and that of the real problem, trust region methods adjust the radius of the trust region to continue to obtain reduction by solving model problems. At the end of this project, it is clear that (1) It is possible to develop a pure bundle method with linear subproblems and without trust region update for convex optimization problems; such method converges to minimizers if it generates an infinite sequence of serious steps; otherwise, it can be shown that the method generates a sequence of minor updates and the last serious step is a minimizer. First, this PhD project develops a bundle trust region algorithm with linear model and linear subproblem for minimizing a prox-regular and Lipschitz function. It adopts a convexification technique from the redistributed bundle method. Global convergence of the algorithm is established in the sense that the sequence of iterations converges to the fixed point of the proximal-point mapping given that convexification is successful. Preliminary numerical tests on standard academic nonsmooth problems show that the algorithm is comparable to bundle methods with quadratic subproblem. Second, following the philosophy behind bundle method of making full use of the previous information of the iteration process and obtaining a flexible understanding of the function structure, the project revises the algorithm developed in the first part by applying the nonmonotone trust region method.We study the performance of numerical implementation and successively refine the algorithm in an effort to improve its practical performance. Such revisions include allowing the convexification parameter to possibly decrease and the algorithm to restart after a finite process determined by various heuristics. The second theme of this project is about the theories of nonsmooth analysis, focusing on U-Lagrangian. When restricted to a subspace, a nonsmooth function can be differentiable within this space. It is known that for a nonsmooth convex function, at a point, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian (called the U-Lagrangian) can be defined and has nice smooth properties and V space, the orthogonal complement subspace of the U space. In this thesis we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function. Similar work in the literature includes a quadratic sub-Lagrangian. It is our interest to study the feasibility of a linear localized U-Lagrangian. We also study the connections of the new U-Lagrangian and other subsmooth structures including fast tracks and partial smooth functions. This part of the project tries to provide answers to the following questions: (1) based on a generalized UV-decomposition, can we develop a linear U-Lagrangian of a prox-regular function that maintains prox-regularity? (2) through the new U-Lagrangian can we show that partial smoothness and fast tracks are equivalent under prox-regularity? At the end of this project, it is clear that for a function f that is properly prox-regular at a point x*, a new linear localized U-Lagrangian can be defined and its value at 0 coincides with f(x*); under some conditions, it can be proved that the U-Lagrangian is also prox-regular at 0; moreover partial smoothness and fast tracks are equivalent under prox-regularity and other mild conditions.
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