Publication Date

2022

Document Type

Dissertation/Thesis

First Advisor

Krislock, Nathan

Degree Name

M.S. (Master of Science)

Legacy Department

Department of Mathematical Sciences

Abstract

\begin{abstract}In this thesis, we concern ourselves with solving the unconstrained optimization problem % \begin{gather*} \text{Minimize}\; f(x)\\\text{subject to}\; x\in X \end{gather*} % where $f\colon\mathbb{R}^N\to \mathbb{R}$ is a non-convex function, possibly with infinitely many local minima. Solving such a problem, especially in higher dimensions often proves to be an extraordinarily difficult task, either in time complexity or in the methodology itself. Indeed, mathematicians must often resort to algorithms which make use of problem structure and which may not generalize well. In this thesis, we present two algorithms which solve this problem, albeit with their own shortcomings.

First, we present a new, $N$-dimensional generalization of an already known deterministic algorithm in $\mathbb{R}$. This new generalization will provide insight into the often difficult space of non-convex optimization and makes use of elegant stopping conditions which provide a sufficient certificate of optimality.

Next, we address the time complexity issues of this algorithm with a known heuristic approach, an approach which is considered best practice among optimization practitioners. While no certificate of optimality may be obtained from this algorithm, its speed and accuracy give it some extreme advantages over deterministic approaches.\end{abstract}

Extent

50 pages

Language

eng

Publisher

Northern Illinois University

Rights Statement

In Copyright

Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

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Text

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