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17
Column Generation
 CONTRIBUTED TO THE WILEY ENCYCLOPEDIA OF OPERATIONS RESEARCH AND MANAGEMENT SCIENCE (EORMS)
, 2010
"... Column generation is an indispensable tool in computational optimization to solve a mathematical program by iteratively adding the variables of the model. Even though the method is simple in theory there are many algorithmic choices and we discuss the most common ones. Particular emphasis in put on ..."
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Cited by 78 (3 self)
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Column generation is an indispensable tool in computational optimization to solve a mathematical program by iteratively adding the variables of the model. Even though the method is simple in theory there are many algorithmic choices and we discuss the most common ones. Particular emphasis in put on the dual interpretation, relating column generation to Langrangian relaxation and cutting plane algorithms, which revealed several critical issues like the need for dual variable stabilization techniques. We conclude with some advise for computer implementations.
Robust branchandcutandprice for the capacitated vehicle routing problem
 IN PROCEEDINGS OF THE INTERNATIONAL NETWORK OPTIMIZATION CONFERENCE
, 2003
"... During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branchandcut algorithms giving bett ..."
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Cited by 60 (15 self)
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During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branchandcut algorithms giving better results. However, several instances in the range of 50–80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over qroutes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branchandcutandprice could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities.
E.: Stabilized branchandcutandprice for the generalized assignment problem
 In: Annals of GRACO’05, Electronic Notes in Discrete Mathematics
, 2005
"... The Generalized Assignment Problem (GAP) is a classic scheduling problem with many applications. We propose a branchandcutandprice for that problem featuring a stabilization mechanism to accelerate column generation convergence. We also propose ellipsoidal cuts, a new way of transforming the exa ..."
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Cited by 8 (1 self)
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The Generalized Assignment Problem (GAP) is a classic scheduling problem with many applications. We propose a branchandcutandprice for that problem featuring a stabilization mechanism to accelerate column generation convergence. We also propose ellipsoidal cuts, a new way of transforming the exact algorithm into a powerful heuristic, in the same spirit of the cuts recently proposed by Fischetti and Lodi. The improved solutions found by this heuristic can, in turn, help the task of the exact algorithm. The resulting algorithms showed a very good performance and were able to solve three among the last five open instances from the ORLibrary. 1
Decomposition and dynamic cut generation in integer linear programming
, 2004
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Robust branchcutandprice for the capacitated minimum spanning tree problem over a large extended formulation
 Universidade Federal Fluminense
, 2008
"... This paper presents a robust branchcutandprice algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to qarbs, a structure that arises from a relaxation of the capacitated prizecollecting arborescence problem in order to make it solvable in pseudopoly ..."
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Cited by 6 (1 self)
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This paper presents a robust branchcutandprice algorithm for the Capacitated Minimum Spanning Tree Problem (CMST). The variables are associated to qarbs, a structure that arises from a relaxation of the capacitated prizecollecting arborescence problem in order to make it solvable in pseudopolynomial time. Traditional inequalities over the arc formulation, like Capacity Cuts, are also used. Moreover, a novel feature is introduced in such kind of algorithms. Powerful new cuts expressed over a very large set of variables could be added, without increasing the complexity of the pricing subproblem or the size of the LPs that are actually solved. Computational results on benchmark instances from the ORLibrary show very significant improvements over previous algorithms. Several open instances could be solved to optimality. 1 1
Uncommon DantzigWolfe reformulation for the temporal knapsack problem
 INFORMS Journal on Computing
"... We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items (as in the classical case), guaranteeing that for each time instant the set of existing selected items has total weight not larger than the kn ..."
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Cited by 5 (2 self)
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We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items (as in the classical case), guaranteeing that for each time instant the set of existing selected items has total weight not larger than the knapsack capacity. We focus on the exact solution of the problem, noting that prior to our work the best method was the straightforward application of a generalpurpose solver to the natural ILP formulation. Our results indicate that much better results can be obtained by using the same generalpurpose solver to tackle a nonstandard DantzigWolfe reformulation in which subproblems are associated with groups of constraints. This is also interesting since the more natural DantzigWolfe reformulation of single constraints performs extremely poorly in practice. 1
Decomposition in Integer Linear Programming
, 2009
"... Both cutting plane methods and traditional decomposition methods are procedures that compute a bound on the optimal value of an integer linear program (ILP) by constructing an approximation to the convex hull of feasible solutions. This approximation is obtained by intersecting the polyhedron associ ..."
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Cited by 2 (1 self)
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Both cutting plane methods and traditional decomposition methods are procedures that compute a bound on the optimal value of an integer linear program (ILP) by constructing an approximation to the convex hull of feasible solutions. This approximation is obtained by intersecting the polyhedron associated with the continuous relaxation, which has an explicit representation, with an implicitly defined polyhedron having a description of exponential size. In this paper, we first review these classical procedures and then introduce a new class of bounding methods called integrated decomposition methods, in which the bound yielded by traditional approaches is potentially improved by introducing a second implicitly defined polyhedron. We also discuss the concept of structured separation, which is related to the wellknown template paradigm for dynamically generating valid inequalities and is central to our algorithmic framework. Finally, we briefly introduce a software framework for implementing the methods discussed in the paper and illustrate the concepts through the presentation of applications. 1