Many people contributed to this research by providing the various examples. I would like to thank Tomas Lozano-Pérez for the image guided surgery work, Edward Wang for the energy minimization example, Fabio Boschetti for the refraction tomography problem, Shanghua Teng for his support on the distant point algorithm, Hal Abelson for the downhill simplex implementation, Rick Lathrop for the reverse folding examples, Silvio Turrini for the work on circuit design, and others who kept this research alive with their interest. Patrick Winston is a great advisor. I must thank him for his continuous support and original ideas. This thesis would not have been written without him. Finally, Michael de la Maza deserves special thanks for his large contribution to this work and for being an excellent research partner.