Effective finite termination procedures in interior-point methods for linear programming

E ective Finite Termination Procedures in Interior-Point Methods for Linear Programming by Pamela Joy Williams Due to the structure of the solution set, an exact solution to a linear program cannot be computed by an interior-point algorithm without adding features, such as nite termination procedures, to the algorithm. Finite termination procedures attempt to compute an exact solution in a nite number of steps. The addition of a nite termination procedure enables interior-point algorithms to generate highly accurate solutions for problems in which the ill-conditioning of the Jacobian in the neighborhood of the solution currently precludes such accuracy. The main ingredients of nite termination are activating the procedure, predicting the optimal partition, formulating a simplemathematicalmodel to compute a solution and developing computational techniques to solve the model. Each of these issues are studied in turn in this thesis. First, the current optimal face identi cation models are extended to bounded variable linear programming problems. Distance to the lower and upper bounds are incorporated into the model to prevent the computed solution from violating the bound constraints. Theory in the literature is extended to the new model. Empirical evidence shows that the proposed model reduces the number of projection attempts needed to nd an exact solution. When early termination is the goal, projection from a pure composite Newton step is advocated. However, the cost may exceed the bene ts due to the average need of more than one projection attempt to nd an exact solution. Variants of Mehrotra's predictor-corrector primal-dual interior-point algorithm provide the foundation for most practical interior-point codes. To take advantage of all available algorithmic information, we investigate the behavior of the Tapia predictor-corrector indicator, which incorporates the corrector step, to identify the optimal partition. Globally, the Tapia predictor-corrector indicator behaves poorly as do all indicators, but locally exhibits fast convergence. Acknowledgments This thesis is dedicated to my parents, W. J. and Jardia Williams, who nurtured my love of mathematics from early childhood until today, encouraged me in the bleakest of hours, gave me wings to y and always welcomed me back with loving arms. My sister, Alethea, deserves a medal for putting up with me while I weathered the ups and downs of graduate school. I am not the easiest person to live with even under the best circumstances. The members of my committee gave generously of their time and energy to mold me into a scientist. For that, I thank them. My senior advisor, Dr. Richard Tapia, provided me with a solid foundation in optimization, especially interior-point methods. He gave mathematics a human face by not only conveying the fundamental concepts but also relaying insights about the person behind the theory or algorithm. I want to thank him for taking me under his wings and for glimpsing my potential as a mathematician. Thanks to Dr. AmrEl-Bakry for his infectious excitement during the course of this research and his willingness to listen to my often muddled explanations of new ideas. His critique of the rough drafts of this thesis and recommended changes improved the readability by a factor of one hundred. I am indebted to Dr. Yin Zhang for his comments and questions about the research. His queries forced me to look at the big picture, justify every statement, and curtail my sloppy tendencies. I thank Dr. David Scott for his patience and understanding. I thank the AT&T Labs Cooperative Research Fellowship Program for its nancial support throughout my graduate career. Special thanks goes to myCRFPmentor, Dr. Patricia Wirth, for sharing her graduate school and professional experiences with me. v It is inspiring to witness her accomplishments rst at AT&T Bell Laboratories and now at AT&T Labs. Moreover, thanks to David Houck and the rest of the Teletra c and Performance Analysis Department for their encouragement throughout the years. The Rice graduate and undergraduate students who have positively impacted my career are too numerous to list here. However, I want to acknowledge the Spend the Summer with a Scientist program (1992-1997) whose participants have provided much needed moral support over the years. Cassandra McZeal, Dr. Anthony Kearsley, and Monica Martinez have all contributed in some fashion to my success. Cassandra McZeal has been a con dante, friend, study partner, and a great source of support over the last six years. Without her assistance in a myriad ways I could not have completed this work. Tony Kearsley was and is a wonderful mentor. From the day I came to visit Rice's campus as an undergraduate to the present, he has dispensed advice about classes and life beyond the hedges. (Tony Because of you, I will always remember Baltimore.) Monica Martinez pushed me to excel in the classroom. Thanks to Theresa Chatman for the use of Macintosh computer resources whenever needed, leftovers from CRPC meetings, and her friendship. I want to thank Daria Lawrence and Fran Moshiri for tracking down my stipend on more than one occasion, answering all my questions no matter how inane, and o ering help with a smile. Linda Neyra made conference travel and graduate life, in general, less complicated. She is a tremendous asset to the Optimization Group and the department. Last but not least, I thank my rst grade teacher, Miss Arletta Bacon, for not sending me home on the many occasions that I feigned illness. Although I resented your tough love then, I now realize you helped de ne who I am today. vi To W.J. and Jardia Williams