Mixed Integer Bilinear Programming with Applications to the Pooling Problem
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MIXED INTEGER BILINEAR PROGRAMMING WITH APPLICATIONS TO THE POOLING PROBLEM A Thesis Presented to The Academic Faculty by Akshay Gupte In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in Operations Research H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology December 2012 MIXED INTEGER BILINEAR PROGRAMMING WITH APPLICATIONS TO THE POOLING PROBLEM Approved by: Professor Shabbir Ahmed, Advisor Associate Professor Joel Sokol H. Milton Stewart School of Industrial H. Milton Stewart School of Industrial and Systems Engineering and Systems Engineering Georgia Institute of Technology Georgia Institute of Technology Asst. Professor Santanu Dey, Advisor Dr. Myun Seok Cheon H. Milton Stewart School of Industrial Corporate Strategic Research and Systems Engineering ExxonMobil Research and Engineering Georgia Institute of Technology Company Professor George Nemhauser Date Approved: August 1, 2012 H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology To my beloved wife Akshi. iii ACKNOWLEDGEMENTS I take this opportunity to thank my advisors, Prof. Shabbir Ahmed and Asst. Prof. Santanu Dey, for their guidance and support throughout my Ph.D. studies. They have provided invaluable advice while conducting research and inspired me to pursue an academic career. Thanks to the rest of my committee, Prof. George Nemhauser, Assoc. Prof. Joel Sokol, and Dr. Myun Seok Cheon for their helpful comments and suggestions. I also appreciate the role of the faculty and staff at the School of Industrial and Systems Engineering in providing me with a high quality education. The seemingly infinite array of courses kept me busy along with my research. Dr. Gary Parker and Ms. Pam Morrison were a great help with administrative issues. Special thanks to my wife, Akshi, for her unconditional love and relentless support during these many years of graduate school. No doubt the journey has been long and arduous, and at times very stressful, however her comforting presence made matters easier for me. I deeply appreciate all the sacrifices she has made to help me achieve my goals. I also thank my parents for presenting me with the opportunity to attend graduate school and for their constant encouragement. I had like to thank my peers at Georgia Tech - Dimitri, Feng, Ahmed, Gustavo, Diego, Pete, Qie, Steve, and many more. Doing research becomes -easier when you have a good support group to exchange ideas and blow off steam outside work. Special mention also goes to the intangibles, such as the Pandoras and Spotifys, whose constant internet streaming enabled writing parts of this document into the wee hours. Finally, I would like to acknowledge the financial support for this research from Exxon- Mobil Corporate Strategic Research. I also thank Dr. Ahmet Keha from ExxonMobil for his guidance during my summer internship. Thanks to School of Industrial and Systems Engineering for John Morris fellowship and graduate assistantships during the initial few semesters. iv TABLE OF CONTENTS DEDICATION ...................................... iii ACKNOWLEDGEMENTS .............................. iv LIST OF TABLES ................................... viii LIST OF FIGURES .................................. ix SUMMARY ........................................ x I POOLING PROBLEM .............................. 1 1.1 Introduction . 1 1.2 Problem Formulations . 3 1.2.1 Model parameters . 4 1.2.2 Concentration model : p-formulation . 6 1.2.3 Alternate formulations . 8 1.3 Problem sizes . 13 1.4 Variants of the pooling problem . 14 1.4.1 Time indexed pooling problem . 15 1.5 Relaxations . 18 1.5.1 Envelopes of bilinear functions . 18 1.5.2 Relaxing feasible sets . 23 1.5.3 Value function and Lagrangian relaxation . 28 1.6 Summary . 33 II BILINEAR SINGLE NODE FLOW ...................... 34 2.1 Introduction . 34 2.2 Basic properties . 38 2.2.1 Relaxing flow conservation . 39 2.2.2 Extreme points . 41 2.3 Standard polyhedral relaxations . 46 2.4 Disjunctive formulation . 52 2.4.1 Restrictions using extreme values . 53 2.4.2 High-dimensional representations . 58 v 2.5 Lifted inequalities . 63 2.5.1 Pairwise sequence independent lifting . 67 2.5.2 Valid inequalities from secants . 75 2.6 Conclusion . 81 2.7 Notes . 82 III MILP APPROACHES TO MIXED INTEGER BILINEAR PROGRAM- MING ......................................... 84 3.1 Introduction . 84 3.2 MILP formulations . 86 3.2.1 Reformulations of single term mixed integer bilinear set . 87 3.2.2 Reformulations of (MIBLP) . 92 3.3 Facets of single term mixed integer bilinear set . 93 3.3.1 Convex hulls of unconstrained bilinear terms . 94 3.3.2 Minimal covers of knapsack . 101 3.3.3 Some extensions . 105 3.4 Computational results . 108 3.4.1 Experimental setup . 108 3.4.2 General mixed integer bilinear problems . 110 3.4.3 Nonconvex objective function with linear constraints . 115 3.5 Bounded bilinear terms . 121 3.5.1 Cut separation . 122 3.5.2 Disjunctive inequalities . 124 3.5.3 Computational results . 129 3.6 General expansion knapsack . 132 3.6.1 Proof of validity . 134 3.6.2 Facets of the convex hull . 137 3.7 Conclusion . 139 IV DISCRETIZATION METHODS FOR POOLING PROBLEM ..... 140 4.1 Variable discretizations . 140 4.1.1 Flow discretization . 144 4.1.2 Ratio and specification discretization . 149 vi 4.2 Network flow MILPs . 150 4.2.1 Discretizing consistency requirements at each pool . 150 4.2.2 Exponentially large formulation for ratio discretization . 154 4.3 Computational results . 156 4.3.1 Experimental setup . 156 4.3.2 Test instances . 158 4.3.3 Preprocessing . 160 4.3.4 Global optimal solutions . 162 4.3.5 MILP results . 163 4.4 Summary . 169 REFERENCES ..................................... 173 vii LIST OF TABLES 1 Comparing problem sizes for alternate formulations of the pooling problem. 13 2 Test instances from MINLPLib . 111 3 Optimality gaps for test instances from MINLPLib . 111 4 Product bundling instances : test set 1. 113 5 Product bundling instances : test set 2. 114 6 The watts instances for product bundling. 115 7 Optimality gaps for watts instances . 115 8 General MILP test instances from MIPLIB . 116 9 Optimality gaps for test instances from MIPLIB . 116 10 BoxQP test instances from [106]. 118 11 Comparing Couenne on two formulations of BoxQP instances. 119 12 Disjoint bilinear instances : test set 1. 120 13 Disjoint bilinear instances : test set 2. 120 14 Cuts from bounded bilinear term : test set 1. 130 15 Cuts from bounded bilinear term : test set 2. 131 16 Cuts from bounded bilinear term : watts instances. 131 17 Nomenclature for bilinear sets. 143 18 Characteristics of the pooling instances. 160 19 Effects of LP preprocessing. 161 20 Global optimal solutions or best upper bounds after time limit. 164 21 Feasible solutions by discretizing outflows from each pool. 167 22 Feasible solutions by discretizing inflow ratios in pq-formulation. 168 23 Discretizing consistency requirements at each pool. 170 24 Summary of discretization methods . 172 viii LIST OF FIGURES 1 A sample pooling problem . 2 2 Single bilinear term and its McCormick envelopes. 19 3 Value functions for instances from Haverly [60]. 31 4 Tracking a single flow component at a node. 35 5 Restrictions of using extremal values. 57 P 6 Comparing restrictions of McCormick relaxations . 63 7 Example of perturbation function . 79 8 Symmetric bounds on variables in (21). 83 9 Enforcing consistency requirements at each pool using a expanded network. 151 10 Expanded network MILP model for ratio discretization . 155 ix SUMMARY This dissertation studies mixed integer bilinear programming (MIBLP) problems, which form a class of optimization problems defined as > > > min x Q0y + f x + g y x;y 0 0 s.t. Ax + Gy h0 ≤ > > > x Qty + f x + g y ht; t = 1; : : : ; p; t t ≤ x; y 0; xi Z+; i ; yj Z+; j : ≥ 2 2 I 2 2 J This problem has two sources of nonconvexity: one due to integer variables and second due to bilinear functions of the form x>Qy + f >x + g>y. Hence, a MIBLP must be solved using a global optimization algorithm to obtain a exact solution. A central theme of this research is the use of mixed integer linear programming (MILP)-type techniques for solving MIBLPs. Mixed integer bilinear programs find many applications, a particular one being the pooling problem. Chapter 1 introduces the pooling problem as a minimum cost network flow problem on a directed graph. The classical pooling problem is a continuous bilinear program (BLP). We review various equivalent optimization models for this problem and also address conventional relaxations obtained by relaxing all the bilinear terms. We compare the strengths of these relaxations and provide stronger than previously-known results about the tightness of the various problem formulations. The main contributions of this dissertation follow in Chapters 2 { 4. In Chapter 2, we investigate relaxations of a single node flow substructure of general network flow problems that involve material mixing phenomenon at certain nodes in the graph. We study some basic properties and generate disjunctive representations from appropriate restrictions of this set. Valid linear inequalities are derived by extending the theory of lifting restrictions of a set from MILP literature. We present explicit expressions for a exponential class of valid inequalities. To the best of our knowledge, this is the first study that contributes a x polyhedral relaxation for this set without adding any auxiliary variables. Chapter 3 studies general MIBLPs where every bilinear term is expressed as the product of one continuous and one integer variable. We propose using a mixed 0; 1 MILP formula- f g tion for solving this problem. Facet-defining inequalities are presented for the convex hull of the MILP reformulation a single mixed integer bilinear term. The proposed cutting planes.