Date of Award
Doctor of Philosophy (PhD)
We apply an interior point algorithm to two nonlinear optimization problems and achieve improved results. We also devise an approximate convex functional alternative for use in one of the problems and estimate its accuracy.
The first problem is maximum variance unfolding in machine learning. The traditional method to solve this problem is to convert it to a semi-definite optimization problem by defining a kernel matrix. We obtain better unfolding and higher speeds with the interior point algorithm on the original non-convex problem for data with less than 10,000 points.
The second problem is a multi-objective dose optimization for intensity modulated radiotherapy, whose goals are to achieve high radiation dose on tumors while sparing normal tissues. Due to tumor motions and patient set-up errors, a robust optimization against motion uncertainties is required to deliver a clinically acceptable treatment plan. The traditional method, to irradiate an enlargement of the tumor region, is very conservative and leads to possibly high radiation dose on sensitive structures. We use a new robust optimization model within the framework of goal programming that consists of multiple optimization steps based on prescription priorities. One metric is defined for each structure of interest. A final robustness optimization step then minimizes the variance of all the goal metrics with respect to the motion probability space, and pushes the mean values of these metrics toward a desired value as well. We show similar high dose coverage on example tumors with reduced dose on sensitive structures.
One clinically important metric for a radiation dose distribution, that describes tumor control probability or normal tissue complication probability, is Dx, the minimum dose value on the hottest x% of a structure. It is not mathematically well-behaved, which impedes its use in optimization. We approximate Dx with a linear function of two generalized equivalent uniform dose metrics, also known as lp norms, requiring that the approximation is concave so that its maximization becomes a convex problem. Results with cross validation on a sampling of radiation therapy plans show that the error of this approximation is less than 1 Gy for the most used range 80 to 95 of x values.
Chair and Committee
Mladen V Wickerhauser
Joseph O Deasy, Yixin Chen, Joseph O Deasy, Renato Feres, Ed Spitznagel
Xie, Yao, "Applications of Nonlinear Optimization" (2014). Arts & Sciences Electronic Theses and Dissertations. 369.