Publication Date


Document Type


First Advisor

Krislock, Nathan

Degree Name

M.S. (Master of Science)

Legacy Department

Department of Mathematical Sciences


\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}


50 pages




Northern Illinois University

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