## Dissertations

Dissertation

8-31-2020

#### Degree Name

Doctor of Philosophy in Computing Sciences - (Ph.D.)

#### Department

Computer Science

James M. Calvin

Craig Gotsman

Marvin K. Nakayama

Zhi Wei

MengChu Zhou

#### Abstract

Global optimization is a classical problem of finding the minimum or maximum value of an objective function. It has applications in many areas, such as biological image analysis, chemistry, mechanical engineering, financial analysis, deep learning and image processing. For practical applications, it is important to understand the efficiency of global optimization algorithms. This dissertation develops and analyzes some new global optimization algorithms and applies them to practical problems, mainly for image registration and data clustering.

First, the dissertation presents a new global optimization algorithm which approximates the optimum using only function values. The basic idea is to use the points at which the function has been evaluated to decompose the domain into a collection of hyper-rectangles. At each step of the algorithm, it chooses a hyper-rectangle according to a certain criterion and the next function evaluation is at the center of the hyper-rectangle. The dissertation includes a proof that the algorithm converges to the global optimum as the number of function evaluations goes to infinity, and also establishes the convergence rate. Standard test functions are used to experimentally evaluate the algorithm.

The second part focuses on applying algorithms from the first part to solve some practical problems. Image processing tasks often require optimizing over some set of parameters. In the image registration problem, one attempts to determine the best transformation for aligning similar images. Such problems typically require minimizing a dissimilarity measure with multiple local minima. The dissertation describes a global optimization algorithm and applies it to the problem of identifying the best transformation for aligning two images.

Global optimization algorithms can also be applied to the data clustering problem. The basic purpose of clustering is to categorize data into different groups by their similarity. The objective cost functions for clustering usually are non-convex. \$k\$-means is a popular algorithm which can find local optima quickly but may not obtain global optima. The different starting points for \$k\$-means can output different local optima. This dissertation describes a global optimization algorithm for approximating the global minimum of the clustering problem.

The third part of the dissertation presents variations of the proposed algorithm that work with different assumptions on the available information, including a version that uses derivatives.

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