LARGE SCALE LINEAR OPTIMIZATION FOR WIRELESS COMMUNICATION SYSTEMS A Thesis Presented in Partial Fulfillment of the Requirements for the Degree Master of Mathematical Science in the Graduate School of The Ohio State University By Sameh Hosny, M.S. Graduate Program in Department of Mathematics The Ohio State University 2017 Master's Examination Committee: Prof. Ghaith Hiary, Advisor Prof. Facundo Memoli © Copyright by Sameh Hosny 2017 Abstract Linear Programming has many applications in the domain of wireless commu- nication. Many problems in this field consist of a very large number of variables and constraints and therefore fit in the platform of large scale linear programming. Advancements in computing over the past decade have allowed us to routinely solve linear programs in thousand of variables and constraints, using specialized methods from large scale linear programming. There are many software packages that im- plement such methods, e.g. AMPL, GAMS and Matlab. This dissertation gives A concise survey of linear programming fundamentals with a focus on techniques for large scale linear programming problems in the context of wireless communication. The dissertation explains some of these techniques, in particular the delayed column generation method and the decomposition method. It also draws on examples from the active field of wireless communication. The dissertation is concluded by giving concrete examples of how to use various software packages to solve large scale lin- ear programming problems stemming from our examples in the context of wireless communication. ii To the soul of my father, to my beloved mother, to my great wife, Doaa Eid and my kids Rinad, Rawan and Mohammed. iii Acknowledgments I would like to express my special appreciation and thanks to my advisor Professor Ghaith Hiary. You have been a tremendous mentor for me. It has been an honor for me to be one of your students. I appreciate all your contributions of time and ideas to make my M.Sc. experience productive and stimulating. The joy and enthusiasm you have for your research was contagious and motivational for me, even during tough times in the M.Sc. pursuit. I am also thankful to all the professors who taught me from the math department. I am really grateful to them all for their dedication and devotion to the courses they educate. These courses helped me create a strong and rigorous background in both my major and minor fields. It allowed me to improve my research skills and to change my perspective to many things. The members of the IPS lab have contributed immensely to my personal and professional time at The Ohio State University. The group has been a source of friendships as well as good advice and collaboration. I would like to acknowledge my colleague John Tadrous for his continuous help and generosity. He was always supporting me with all the information I needed especially in the beginning of my study. Moreover, I am thankful to my colleague Faisal Alotaibi for the great time we spent working together and having useful technical discussions in our group meetings. My time at OSU was made enjoyable in large part due to the many friends and groups that became a part of my life. I would like to extend my special thanks to my iv best Egyptian friend Sameh Shohdy and his great family for their kindness, support, and hospitality. They supported me and my family until everything was settle down in Columbus. I also experess my thanks to our great American firends, Betty Rocke and Randy, for supporting our stay in Columbus and helping my son Mohammed in learning so many things. Lastly, I would like to thank my family for all their love and encouragement. For my parents who raised me with a love of science and supported me in all my pursuits. And most of all for my loving, supportive, encouraging, and patient wife Doaa Eid whose faithful support during all stages of this Ph.D. is so appreciated. For spending many nights waiting for me to accomplish my hard tasks. Thank you. v Vita December 11, 1978 . Born - Cairo, Egypt 2001 . .B.S. Electrical and Computer Engi- neering 2010 . .M.S. Electrical and Computer Engi- neering 2014-present . .Ph.D. Student, Electrical and Com- puter Engineering, The Ohio State University 2015-present . .M.S. Student, Mathematics, The Ohio State University Publications (Accepted) S. Hosny, A. Eryilmaz and H. El Gamal, "Impact of User Mobility on D2D Caching Networks," IEEE Global Communications Conference, Washignton DC, USA, 2016. (Accepted) S. Hosny, A. Eryilmaz and H. El Gamal, "Mobility-Aware Centralized D2D Caching Networks," 54th Annual Allerton Conference on Communication, Con- trol, and Computing, Illinois, USA, 2016. (Submitted) S. Hosny, F. Alotaibi, J. Tadrous, A. Eryilmaz and H. El Gamal, "Con- tent Trading in D2D Caching Networks," IEEE/ACM Transactions on Networking. (To be submitted) S. Hosny, A. Abouzeid, A. Eryilmaz and H. El Gamal, "Mobility- Aware D2D Caching Networks," IEEE Transactions on Wireless Communications. F. Alotaibi, S. Hosny, H. El Gamal and A. Eryilmaz, "A game theoretic approach to content trading in proactive wireless networks," 2015 IEEE International Symposium on Information Theory (ISIT), Hong Kong, 2015, pp. 2216-2220. vi S. Hosny, F. Alotaibi, H. E. Gamal and A. Eryilmaz, "Towards a P2P mobile contents trading," 2015 49th Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA, 2015, pp. 338-342. S. Hosny, F. Alotaibi, H. El Gamal and A. Eryilmaz, "Towards a mobile content marketplace," 2015 IEEE 16th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Stockholm, 2015, pp. 675-679. Alotaibi, F., S. Hosny, J. Tadrous, H. El Gamal, and A. Eryilmaz. "Towards a mar- ketplace for mobile content: Dynamic pricing and proactive caching." arXiv preprint arXiv:1511.07573 (2015). Fields of Study Major Field: Electrical & Computer Engineering vii Table of Contents Page Abstract . ii Dedication . iii Acknowledgments . iv Vita......................................... vi List of Tables . x List of Figures . xi 1. Introduction . 1 2. Review of Linear Programming . 3 2.1 Introduction . 3 2.2 Geometry of a Linear Program . 7 2.3 Degeneracy . 12 2.4 The Simplex Method . 13 2.4.1 Implementation of the Simplex Method . 16 2.4.2 Comparisons and Performance Enhancements . 20 2.5 The Duality Theory . 24 2.6 Example Problems for Wireless Communication Networks . 27 2.6.1 Power Control in a Wireless Network . 27 2.6.2 Multicommodity Network Flow . 28 2.6.3 D2D Caching Networks . 30 viii 3. Large Scale Linear Programs . 33 3.1 Delayed Column Generation Method . 34 3.2 Cutting Plane Method . 38 3.3 Dantzig-Wolfe Decomposition . 40 3.4 The Cutting Stock Problem . 45 3.5 Applications in Wireless Communication . 48 4. Implementation of Large Scale Linear Programs . 51 4.1 AMPL Programming Language . 52 4.1.1 Implementation of The Cutting Stock Problem using AMPL 53 4.2 GAMS Programming Language . 57 4.2.1 Implementation of Dantzig-Wolfe Decomposition Method us- ing GAMS . 57 4.3 Matlab Programming Language . 62 4.3.1 Matlab Implementation of D2D Caching Example . 64 Appendices 68 A. AMPL Implementation of Column Generation . 68 B. GAMS Implementation of Multi-Commodity Network Flow Problem . 71 C. Matlab Implementation of D2D Caching Example . 75 Bibliography . 79 ix List of Tables Table Page 2.1 Comparison between Simplex implementation methods . 22 2.2 The Different Possibilities for the Primal and Dual Problems . 26 x List of Figures Figure Page 2.1 Graphical solution of a linear program example. .8 2.2 Visualization of standard form problems . .8 2.3 Full Tableau Structure . 19 2.4 An illustration of the power control example. 28 2.5 An illustration of the multi-commodity network flow example. 30 2.6 An illustration of the D2D caching networks example. 32 4.1 System Performance of the D2D Caching Network . 67 xi Chapter 1: Introduction The importance of Linear Programming (LP) derives in part from its many ap- plications and in part from the existence of efficient techniques to solve it. These techniques are fast and reliable over a substantial range of problem sizes, inputs and applications. Linear programming has been proven to be valuable for modeling di- verse types of problems in planning, routing, scheduling, assignment, and design. Industries that make use of LP and its extensions include transportation, energy, telecommunications, health care, finance and manufacturing. In a number of these applications, a realistic model gives rise to a LP problem with a large number of variables and constraints. This makes the problem more complicated and requires substantial computational resources to solve it; in particular, substantial amount of fast memory and higher computational speed. For this reason, a number of special- ized procedures, such as column generation and cutting-plane methods, have been developed to effectively solve such large-scale linear programs. Yet, in other cases, the LP problem may have a special structure where the decomposition methods can be useful. Linear programming has numerous and important applications in the domain of wireless communications, e.g. network flow, power control, caching networks, etc. Most of these applications deal with very large number of variables and constraints. 1 For example, caching networks deal with a very large number of users and a tremen- dous amount of data contents. Therefore, we focus in this dissertation on linear programming methods for such large scale problems. We also investigate some soft- ware packages to implement and solve these problems. Thanks to the advances in computing over the past decade, linear programs in a few thousand variables and constraints are nowadays viewed as "small" problems. Problems having tens, or even hundreds, of thousands of continuous variables are regularly solved using software packages such as AMPL, GAMS, Matlab, etc.