Date of Award

Winter 12-17-2019

Level of Access

Open-Access Thesis

Degree Name

Master of Arts (MA)

Department

Mathematics

Advisor

Peter Stechlinski

Second Committee Member

Andrew Knightly

Third Committee Member

David Bradley

Abstract

Derivative information is useful for many problems found in science and engineering that require equation solving or optimization. Driven by its utility and mathematical curiosity, researchers over the years have developed a variety of generalized derivatives. In this thesis, we will first take a look at Clarke’s generalized derivative for locally Lipschitz continuous functions between Euclidean spaces, which roughly is the smallest convex set containing all nearby derivatives of a domain point of interest. Clarke’s generalized derivative in this setting possesses a strong theoretical and numerical toolkit, which is analogous to that of the classical derivative. It includes nonsmooth versions of the chain rule, the mean value theorem, and the implicit function theorem, as well as nonsmooth equation-solving and optimization methods. However, it is generally difficult to obtain elements of Clarke’s generalized derivative in the Euclidean space setting. To address this issue, we use lexicographic differentiation by Nesterov and lexicographic directional differentiation by Khan and Barton. They are generalized derivatives theories for a subclass of locally Lipschitz continuous functions, called the class of lexicographically smooth functions, which help to find elements of Clarke's generalized derivative in the Euclidean space setting systematically. Lexicographic derivatives are either elements of Clarke's generalized derivative in the Euclidean space setting or at least indistinguishable from them as far as numerical tools are concerned. We outline a process by which we can find a lexicographic derivative once a lexicographic directional derivative is known. Lastly, we present lexicographic differentiation theory for a subclass of locally Lipschitz continuous functions mapping between Banach spaces that have Schauder bases, called, unsurprisingly, the class of lexicographically smooth functions. We provide a proof for Nesterov's result that, as in the Euclidean space setting, lexicographic derivatives in this setting satisfy a sharp calculus rule.

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