University of Tennessee, Knoxville TRACE: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2014 Indefinite napsackK Separable Quadratic Programming: Methods and Applications Jaehwan Jeong University of Tennessee - Knoxville, [email protected] Follow this and additional works at: https://trace.tennessee.edu/utk_graddiss Part of the Finance and Financial Management Commons, Management Sciences and Quantitative Methods Commons, Other Mathematics Commons, Portfolio and Security Analysis Commons, and the Theory and Algorithms Commons Recommended Citation Jeong, Jaehwan, "Indefinite napsackK Separable Quadratic Programming: Methods and Applications. " PhD diss., University of Tennessee, 2014. https://trace.tennessee.edu/utk_graddiss/2704 This Dissertation is brought to you for free and open access by the Graduate School at TRACE: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of TRACE: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Jaehwan Jeong entitled "Indefinite napsackK Separable Quadratic Programming: Methods and Applications." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the equirr ements for the degree of Doctor of Philosophy, with a major in Management Science. Chanaka Edirisinghe, Major Professor We have read this dissertation and recommend its acceptance: Hamparsum Bozdogan, Bogdan Bichescu, James Ostrowski Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official studentecor r ds.) University of Tennessee, Knoxville Trace: Tennessee Research and Creative Exchange Doctoral Dissertations Graduate School 5-2014 Indefinite Knapsack Separable Quadratic Programming: Methods and Applications Jaehwan Jeong University of Tennessee - Knoxville, [email protected] This Dissertation is brought to you for free and open access by the Graduate School at Trace: Tennessee Research and Creative Exchange. It has been accepted for inclusion in Doctoral Dissertations by an authorized administrator of Trace: Tennessee Research and Creative Exchange. For more information, please contact [email protected]. To the Graduate Council: I am submitting herewith a dissertation written by Jaehwan Jeong entitled "Indefinite Knapsack Separable Quadratic Programming: Methods and Applications." I have examined the final electronic copy of this dissertation for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of Doctor of Philosophy, with a major in Management Science. Chanaka Edirisinghe, Major Professor We have read this dissertation and recommend its acceptance: Hamparsum Bozdogan, Bogdan Bichescu, James Ostrowski Accepted for the Council: Carolyn R. Hodges Vice Provost and Dean of the Graduate School (Original signatures are on file with official student records.) Indefinite Knapsack Separable Quadratic Programming: Methods and Applications A Dissertation Presented for the Doctor of Philosophy Degree The University of Tennessee, Knoxville Jaehwan Jeong May 2014 Copyright ©2014 by Jaehwan Jeong All rights reserved. ii Dedicated to My Parents, Sister, Brother and My lovely Wife iii Acknowledgement First and foremost, I would like to express my best gratitude to my advisor, Dr. Chanaka Edirisinghe. There is no doubt that he poured his heart into me. I am sure that I could not reach this professional level of the study without his sincere consideration and passion. He has the distinguished ability to train a student as a scientific thinker. When I first met him, I was like a wild horse that solely depends on its instinct and crude knowledge. He has patiently sharpened my thinking process and continuously showed his brilliant approaches toward research. Moreover, he has steadily demonstrated how a person can devote enthu- siasm and professionalism for one’s life. He also emphasizes cultivating not only excellent researchers but also “right” researchers who purse moral principles with ethical philosophy. I am really honored to have him as my advisor. He absolutely deserves to receive my full respect. If my future research adds any value for human beings, all the origin sources are from him. I also want to give my special thanks to my dissertation committee members, Dr. Ham- parsum Bozdogan, Dr. Bogdan Bichescu, and Dr. James Ostrowski. Their gentle hearts always cheered me up, and their luminous and explicit advice has guided me to increase the value of my research and life. My graduate study has always been enjoyable being with them, and their profound attitude for life and research are the role models to design my life. I am grateful to my parents who initiated my dream for research. Their invaluable en- couragements, endless supports, and unconditional love are the fundamental foundation of this dissertation. My sister and brother have also been constant sources of encouragement and wishes. This dissertation could not have been accomplished without the dedication, support, and love of my beautiful wife Jungim Yoo. Her strong encouragement has been always the main source to keep me going forward. Her silent patience is definitely the substantial part of this dissertation. I love yoo (you)! Lastly, the words in this acknowledgement would never be able to express my gratitude to people who supported me. This dissertation is just a piece of proof of their love. iv Abstract Quadratic programming (QP) has received significant consideration due to an extensive list of applications. Although polynomial time algorithms for the convex case have been devel- oped, the solution of large scale QPs is challenging due to the computer memory and speed limitations. Moreover, if the QP is nonconvex or includes integer variables, the problem is NP-hard. Therefore, no known algorithm can solve such QPs efficiently. Alternatively, row-aggregation and diagonalization techniques have been developed to solve QP by a sub- problem, knapsack separable QP (KSQP), which has a separable objective function and is constrained by a single knapsack linear constraint and box constraints. KSQP can therefore be considered as a fundamental building-block to solve the general QP and is an important class of problems for research. For the convex KSQP, linear time algorithms are available. However, if some quadratic terms or even only one term is negative in KSQP, the problem is known to be NP-hard, i.e. it is notoriously difficult to solve. The main objective of this dissertation is to develop efficient algorithms to solve general KSQP. Thus, the contributions of this dissertation are five-fold. First, this dissertation includes comprehensive literature review for convex and nonconvex KSQP by considering their computational efficiencies and theoretical complexities. Second, a new algorithm with quadratic time worst-case complexity is developed to globally solve the nonconvex KSQP, having open box constraints. Third, the latter global solver is utilized to develop a new bounding algorithm for general KSQP. Fourth, another new algorithm is developed to find a bound for general KSQP in linear time complexity. Fifth, a list of comprehensive applications for convex KSQP is introduced, and direct applications of indefinite KSQP are described and tested with our newly developed methods. Experiments are conducted to compare the performance of the developed algorithms with that of local, global, and commercial solvers such as IBM Cplex using randomly generated problems in the context of certain applications. The experimental results show that our proposed methods are superior in speed as well as in the quality of solutions. v Table of Contents 1. Introduction1 1.1. Motivations for KSQP...............................5 1.2. Research Scope...................................6 1.3. Contributions of this Dissertation.........................6 1.4. Outline of this Dissertation............................8 1.5. Notation and Preliminaries............................9 2. Knapsack Separable Quadratic Programming (KSQP): Survey 13 2.1. Introduction..................................... 13 2.2. Strictly Convex case................................ 16 2.3. Positive Semidefinite case............................. 53 2.4. Nonconvex case................................... 54 2.5. Conclusion...................................... 61 3. Indefinite Open-Box Approach to Solution 62 3.1. Introduction..................................... 62 3.2. Open-Box Global (OBG) Solver........................... 63 3.3. Bounding with Open-Box Global (OBG) solver.................. 85 3.4. Experiments..................................... 89 3.5. Conclusion...................................... 97 4. Lagrangian-relaxed Closed-Box Solution 99 4.1. Introduction..................................... 99 4.2. Closed-Box Solver (CBS).............................. 100 4.3. Methodology and Implementation......................... 123 4.4. Experiments..................................... 131 4.5. Conclusion..................................... 162 5. Applications 165 5.1. Applications for Convex case........................... 165 5.2. Applications for Nonconvex case......................... 167 6. Concluding Remarks 175 Reference 179 Appendix 195 A. Appendix 196 A.1. The case of aj = 0 ................................