Advances in interior point methods and column generation

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 unstable 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 techniques, 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 solutions are used in the course of the algorithm. Additionally, we present a comprehensive computational study of the solution of linear relaxed formulations obtained after applying 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 speci c 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 warmstarting 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 this thesis restores primal and dual feasibilities after the addition of new columns in one step. The direction is determined so that the modi cation 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

Tactical Problems in Vehicle Routing Applications

The class of Vehicle Routing Problems (VRPs) is one the most studied topics in the Operations Research community. The vast majority of the published papers focus on single-period problems, with a few branches of the literature considering multiperiod generalisations. All of these problems though, consider a short horizon and aim at optimising the decisions at an operational level, i.e. that will have to be taken in the near future. One step above are tactical problems, i.e. problems concerning a longer time horizon. Tactical problems are of a fundamental importance as they directly influence the daily operations, and therefore a part of the incurred costs, for a long time. The main focus of this thesis is to study tactical problems arising in routing applications. The first problem considered concerns the design of a fleet of vehicles. Transportation providers often have to design a fleet that will be used for daily operations across a long-time span. Trucks used for transportation are very expensive to purchase, maintain or hire. On the other side, the composition of the fleet strongly influences the daily plans, and therefore costs such as fuel or drivers’ wages. Balancing these two components is challenging, and optimisation models can lead to substantial savings or provide a useful basis for informed decisions. The second problem presented focuses on the use of a split deliveries policy in multi-period routing problems. It is known that the combined optimisation of delivery scheduling and routing can be very beneficial, and lead to significant reductions in costs. However, it also adds complexity to the model. The same is true when split deliveries are introduced. The problem studied considers the possibility of splitting the deliveries over different days. An analysis, both theoretical and numerical, of the impact of this approach on the overall cost is provided. Finally, a districting problem for routing applications is considered. These types of problems typically arise when transportation providers wish to increase their service consistency. There are several reasons a company may wish to do so: to strengthen the customer-driver relationship, to increase drivers’ familiarity with their service area, or, to simplify the management of the service area. A typical approach, considered here, is to divide the area under consideration in sectors that will be subsequently assigned to specific drivers. This type of problem is inherently of a multi-period and tactical nature. A new formulation is proposed, integrating standard routing models into the design of territories. This makes it possible to investigate how operational constraints and other requirements, such as having a fair workload division amongst drivers, influence the effectiveness of the approach. An analysis of the cost of districting, in terms of increased routing cost and decreased routing flexibility, and of several operational constraints, is presented

On High-Performance Benders-Decomposition-Based Exact Methods with Application to Mixed-Integer and Stochastic Problems

RÉSUMÉ : La programmation stochastique en nombres entiers (SIP) combine la difficulté de l’incertitude et de la non-convexité et constitue une catégorie de problèmes extrêmement difficiles à résoudre. La résolution efficace des problèmes SIP est d’une grande importance en raison de leur vaste applicabilité. Par conséquent, l’intérêt principal de cette dissertation porte sur les méthodes de résolution pour les SIP. Nous considérons les SIP en deux étapes et présentons plusieurs algorithmes de décomposition améliorés pour les résoudre. Notre objectif principal est de développer de nouveaux schémas de décomposition et plusieurs techniques pour améliorer les méthodes de décomposition classiques, pouvant conduire à résoudre optimalement divers problèmes SIP. Dans le premier essai de cette thèse, nous présentons une revue de littérature actualisée sur l’algorithme de décomposition de Benders. Nous fournissons une taxonomie des améliorations algorithmiques et des stratégies d’accélération de cet algorithme pour synthétiser la littérature et pour identifier les lacunes, les tendances et les directions de recherche potentielles. En outre, nous discutons de l’utilisation de la décomposition de Benders pour développer une (méta- )heuristique efficace, décrire les limites de l’algorithme classique et présenter des extensions permettant son application à un plus large éventail de problèmes. Ensuite, nous développons diverses techniques pour surmonter plusieurs des principaux inconvénients de l’algorithme de décomposition de Benders. Nous proposons l’utilisation de plans de coupe, de décomposition partielle, d’heuristiques, de coupes plus fortes, de réductions et de stratégies de démarrage à chaud pour pallier les difficultés numériques dues aux instabilités, aux inefficacités primales, aux faibles coupes d’optimalité ou de réalisabilité, et à la faible relaxation linéaire. Nous testons les stratégies proposées sur des instances de référence de problèmes de conception de réseau stochastique. Des expériences numériques illustrent l’efficacité des techniques proposées. Dans le troisième essai de cette thèse, nous proposons une nouvelle approche de décomposition appelée méthode de décomposition primale-duale. Le développement de cette méthode est fondé sur une reformulation spécifique des sous-problèmes de Benders, où des copies locales des variables maîtresses sont introduites, puis relâchées dans la fonction objective. Nous montrons que la méthode proposée atténue significativement les inefficacités primales et duales de la méthode de décomposition de Benders et qu’elle est étroitement liée à la méthode de décomposition duale lagrangienne. Les résultats de calcul sur divers problèmes SIP montrent la supériorité de cette méthode par rapport aux méthodes classiques de décomposition. Enfin, nous étudions la parallélisation de la méthode de décomposition de Benders pour étendre ses performances numériques à des instances plus larges des problèmes SIP. Les variantes parallèles disponibles de cette méthode appliquent une synchronisation rigide entre les processeurs maître et esclave. De ce fait, elles souffrent d’un important déséquilibre de charge lorsqu’elles sont appliquées aux problèmes SIP. Cela est dû à un problème maître difficile qui provoque un important déséquilibre entre processeur et charge de travail. Nous proposons une méthode Benders parallèle asynchrone dans un cadre de type branche-et-coupe. L’assouplissement des exigences de synchronisation entraine des problèmes de convergence et d’efficacité divers auxquels nous répondons en introduisant plusieurs techniques d’accélération et de recherche. Les résultats indiquent que notre algorithme atteint des taux d’accélération plus élevés que les méthodes synchronisées conventionnelles et qu’il est plus rapide de plusieurs ordres de grandeur que CPLEX 12.7.----------ABSTRACT : Stochastic integer programming (SIP) combines the difficulty of uncertainty and non-convexity, and constitutes a class of extremely challenging problems to solve. Efficiently solving SIP problems is of high importance due to their vast applicability. Therefore, the primary focus of this dissertation is on solution methods for SIPs. We consider two-stage SIPs and present several enhanced decomposition algorithms for solving them. Our main goal is to develop new decomposition schemes and several acceleration techniques to enhance the classical decomposition methods, which can lead to efficiently solving various SIP problems to optimality. In the first essay of this dissertation, we present a state-of-the-art survey of the Benders decomposition algorithm. We provide a taxonomy of the algorithmic enhancements and the acceleration strategies of this algorithm to synthesize the literature, and to identify shortcomings, trends and potential research directions. In addition, we discuss the use of Benders decomposition to develop efficient (meta-)heuristics, describe the limitations of the classical algorithm, and present extensions enabling its application to a broader range of problems. Next, we develop various techniques to overcome some of the main shortfalls of the Benders decomposition algorithm. We propose the use of cutting planes, partial decomposition, heuristics, stronger cuts, and warm-start strategies to alleviate the numerical challenges arising from instabilities, primal inefficiencies, weak optimality/feasibility cuts, and weak linear relaxation. We test the proposed strategies with benchmark instances from stochastic network design problems. Numerical experiments illustrate the computational efficiency of the proposed techniques. In the third essay of this dissertation, we propose a new and high-performance decomposition approach, called Benders dual decomposition method. The development of this method is based on a specific reformulation of the Benders subproblems, where local copies of the master variables are introduced and then priced out into the objective function. We show that the proposed method significantly alleviates the primal and dual shortfalls of the Benders decomposition method and it is closely related to the Lagrangian dual decomposition method. Computational results on various SIP problems show the superiority of this method compared to the classical decomposition methods as well as CPLEX 12.7. Finally, we study parallelization of the Benders decomposition method. The available parallel variants of this method implement a rigid synchronization among the master and slave processors. Thus, it suffers from significant load imbalance when applied to the SIP problems. This is mainly due to having a hard mixed-integer master problem that can take hours to be optimized. We thus propose an asynchronous parallel Benders method in a branchand- cut framework. However, relaxing the synchronization requirements entails convergence and various efficiency problems which we address them by introducing several acceleration techniques and search strategies. In particular, we propose the use of artificial subproblems, cut generation, cut aggregation, cut management, and cut propagation. The results indicate that our algorithm reaches higher speedup rates compared to the conventional synchronized methods and it is several orders of magnitude faster than CPLEX 12.7

"Rotterdam econometrics": publications of the econometric institute 1956-2005

This paper contains a list of all publications over the period 1956-2005, as reported in the Rotterdam Econometric Institute Reprint series during 1957-2005.

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order