Advances in Interior Point Methods and Column Generation Pablo Gonz´alezBrevis Doctor of Philosophy University of Edinburgh 2013 Declaration I declare that this thesis was composed by myself and that the work contained therein is my own, except where explicitly stated otherwise in the text. (Pablo Gonz´alezBrevis) ii To my endless inspiration and angels on earth: Paulina, Crist´obal and Esteban iii Abstract In this thesis we study how to efficiently combine the column generation technique (CG) and interior point methods (IPMs) for solving the relaxation of a selection of integer programming problems. In order to obtain an efficient method a change in the column generation technique and a new reoptimization strategy for a primal-dual interior point method are proposed. It is well-known that the standard column generation technique suffers from un- stable behaviour due to the use of optimal dual solutions that are extreme points of the restricted master problem (RMP). This unstable behaviour slows down column generation so variations of the standard technique which rely on interior points of the dual feasible set of the RMP have been proposed in the literature. Among these tech- niques, there is the primal-dual column generation method (PDCGM) which relies on sub-optimal and well-centred dual solutions. This technique dynamically adjusts the column generation tolerance as the method approaches optimality. Also, it relies on the notion of the symmetric neighbourhood of the central path so sub-optimal and well-centred solutions are obtained. We provide a thorough theoretical analysis that guarantees the convergence of the primal-dual approach even though sub-optimal solu- tions are used in the course of the algorithm. Additionally, we present a comprehensive computational study of the solution of linear relaxed formulations obtained after apply- ing the Dantzig-Wolfe decomposition principle to the cutting stock problem (CSP), the vehicle routing problem with time windows (VRPTW), and the capacitated lot sizing problem with setup times (CLSPST). We compare the performance of the PDCGM with the standard column generation method (SCGM) and the analytic centre cutting planning method (ACCPM). Overall, the PDCGM achieves the best performance when compared to the SCGM and the ACCPM when solving challenging instances from a column generation perspective. One important characteristic of this column generation strategy is that no specific tuning is necessary and the algorithm poses the same level of difficulty as standard column generation method. The natural stabilization available in the PDCGM due to the use of sub-optimal well-centred interior point solutions is a very attractive feature of this method. Moreover, the larger the instance, the better is the relative performance of the PDCGM in terms of column generation iterations and CPU time. The second part of this thesis is concerned with the development of a new warmstart- ing strategy for the PDCGM. It is well known that taking advantage of the previously solved RMP could lead to important savings in solving the modified RMP. However, this is still an open question for applications arising in an integer optimization context and the PDCGM. Despite the current warmstarting strategy in the PDCGM working well in practice, it does not guarantee full feasibility restorations nor considers the quality of the warmstarted iterate after new columns are added. The main motivation of the design of the new warmstarting strategy presented in this thesis is to close this theoretical gap. Under suitable assumptions, the warmstarting procedure proposed in iv this thesis restores primal and dual feasibilities after the addition of new columns in one step. The direction is determined so that the modification of small components at a particular solution is not large. Additionally, the strategy enables control over the new duality gap by considering an expanded symmetric neighbourhood of the central path. As observed from our computational experiments solving CSP and VRPTW, one can conclude that the warmstarting strategies for the PDCGM are useful when dense columns are added to the RMP (CSP), since they consistently reduce the CPU time and also the number of iterations required to solve the RMPs on average. On the other hand, when sparse columns are added (VRPTW), the coldstart used by the interior point solver HOPDM becomes very efficient so warmstarting does not make the task of solving the RMPs any easier. v Acknowledgements I would like to thank Jacek Gondzio for being such a wonderful supervisor. I have learnt a lot from him and for that I am eternally grateful. I would like to express my gratitude to my examiners, Julian Hall and John Mitchell for their insightful comments and suggestions about this work. I would like to thank Universidad del Desarrollo, particularly the Research Council and the School of Engineering, for their financial support. Also, I am grateful to MIDEPLAN and CONICYT for granting me the Presidente de la Rep´ublica Scholarship. I would like to thank the School of Mathematics at The University of Edinburgh for its financial support during the last six months of my studies. I am also very grateful to everyone who supported me and my family during this \adventure". Specially, I am indebted to Nora Au and Pedro Silva for their constant support from Chile since the beginning. I would also like to thank all my office mates, Bubacarr, Naiyuan, Kimonas, Robert, Tim, Hanyi and Luciana. It was great sharing room 5620 with you guys. Special thanks go to Pedro, Robert and Tim for proofreading a first draft of this thesis. I am also indebted to the staff and administrators at the School of Mathematics at The University of Edinburgh. Particularly, friends and colleges at the Edinburgh Research Group on Optimization (ERGO). I would like to thank all baristas at the Starbucks store at Forrest Road. You always made my mornings (and some afternoons) with a delicious mocha. In a more personal note, I am very grateful to friends I had the chance to meet while in Edinburgh. To Cristian, Loreto, Bryan, Lorena, Pedro, H´ector,Valeria and Milan, you are wonderful people! I would like to thank my parents, Ver´onicaand Enrique; my brothers and sister, Claudio, Sebasti´anand Patricia; and my parents-in-law, Carolina and Julio, for their constant support from far away. You have always believed in me! Last but not least, this thesis could not have been completed without the support of my family. To Crist´obaland Esteban for showing me the beauty on simple things. To Paulina, for her constant encouragement, endless patience and infinite love! A todos, gracias totales! vi Contents Abstract v 1 Introduction 1 1.1 Linear programming problems . .1 1.2 Motivation and contributions . .2 2 Primal-Dual Interior Point Methods 8 2.1 Neighbourhoods and path-following methods . 11 2.2 Convergence and complexity of the Ns(γ) neighbourhood . 14 2.2.1 Algorithm . 14 2.2.2 Convergence . 14 2.2.3 Complexity . 16 3 Column Generation and the Decomposition Principle 17 3.1 Dantzig-Wolfe decomposition . 17 3.2 Lagrangian relaxation equivalence . 20 3.3 A graphical interpretation . 21 3.4 Column generation . 23 3.4.1 Adding columns to the RMP . 26 3.4.2 Limitations of the standard version . 27 3.5 Stabilization strategies . 29 4 A Selection of Problems: Formulation and Decomposition 32 4.1 Cutting stock problem . 32 4.1.1 Dantzig-Wolfe decomposition . 33 4.2 Vehicle routing problem with time windows . 34 4.2.1 Dantzig-Wolfe decomposition . 35 4.3 Capacitated lot-sizing problem with setup times . 36 4.3.1 Dantzig-Wolfe decomposition . 37 5 Primal-Dual Column Generation Method 39 5.1 Theoretical developments . 40 5.2 Computational study . 46 5.2.1 Cutting stock problem . 48 5.2.2 Vehicle routing problem with time windows . 50 5.2.3 Capacitated lot-sizing problem with setup times . 54 5.2.4 Performance profiles for large instances . 56 5.2.5 Additional comments about stabilized column generation . 60 vii 6 A New Reoptimization Technique for the Primal-Dual Column Gen- eration Method 62 6.1 Warmstarting strategies . 63 6.2 Theoretical developments . 65 6.2.1 Technical conditions . 68 6.2.2 Dual feasibility restoration . 70 6.2.3 Primal feasibility restoration . 73 6.2.4 Centrality . 76 6.2.5 Algorithm . 79 6.3 Computational study . 81 6.3.1 Cutting stock problem . 82 6.3.2 Vehicle routing problem with time windows . 86 7 Conclusions and Further Developments 91 7.1 Summary and concluding remarks . 91 7.2 Further studies . 93 A Test problems 96 A.1 Cutting stock problem . 96 A.2 Vehicle routing problem with time windows . 100 A.3 Capacitated lot-sizing problem with setup times . 102 viii List of Tables 5.1 Average results on 262 instances of CSP for the SCGM, the PDCGM and the ACCPM adding one column at a time. 49 5.2 Average results on 262 instances of CSP for the SCGM, the PDCGM and the ACCPM adding up to k columns at a time. 49 5.3 Results on 14 large instances of CSP for the SCGM, the PDCGM and the ACCPM adding up to 100 columns at a time. 50 5.4 Average results on 87 instances of VRPTW for the SCGM, the PDCGM and the ACCPM adding one column at a time. 52 5.5 Average results on 87 instances of VRPTW for the SCGM, the PDCGM and the ACCPM adding up to k columns at a time. 53 5.6 Results on 9 large instances of VRPTW for the SCGM, the PDCGM and the ACCPM adding 300 columns at a time. 54 5.7 Average results on 751 instances of CLSPST for the SCGM, the PDCGM and the ACCPM adding one column per subproblem at a time.