Portland State University. Department of Mathematics and Statistics
Mau Nam Nguyen
Term of Graduation
Date of Publication
Doctor of Philosophy (Ph.D.) in Mathematical Sciences
Mathematics and Statistics
Convex domains, Mathematical optimization, Calculus
1 online resource (vi, 116 pages)
This thesis contains contributions in two main areas: calculus rules for generalized differentiation and optimization methods for solving nonsmooth nonconvex problems with applications to multifacility location and clustering. A variational geometric approach is used for developing calculus rules for subgradients and Fenchel conjugates of convex functions that are not necessarily differentiable in locally convex topological and Banach spaces. These calculus rules are useful for further applications to nonsmooth optimization from both theoretical and numerical aspects. Next, we consider optimization methods for solving nonsmooth optimization problems in which the objective functions are not necessarily convex. We particularly focus on the class of functions representable as differences of convex functions. This class of functions is broad enough to cover many problems in facility location and clustering, while the generalized differentiation tools from convex analysis can be applied. We develop algorithms for solving a number of multifacility location and clustering problems and computationally implement these algorithms via MATLAB. The methods used throughout this thesis involve DC programming, Nesterov's smoothing technique, and the DCA, a numerical algorithm for minimizing differences of convex functions to cope with the nonsmoothness and nonconvexity.
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Tran, Tuyen Dang Thanh, "Convex and Nonconvex Optimization Techniques for Multifacility Location and Clustering" (2020). Dissertations and Theses. Paper 5482.