First Advisor

Mau Nam Nguyen

Term of Graduation

Spring 2020

Date of Publication

6-2-2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy (Ph.D.) in Mathematical Sciences

Department

Mathematics and Statistics

Language

English

Subjects

Convex domains, Mathematical optimization, Calculus

Physical Description

1 online resource (vi, 116 pages)

Abstract

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.

Rights

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Persistent Identifier

https://archives.pdx.edu/ds/psu/33318

Included in

Mathematics Commons

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