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Partial convexification of general MIPs by DantzigWolfe reformulation
 INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION, VOLUME 6655 OF LECT. NOTES COMPUT. SCI
, 2011
"... DantzigWolfe decomposition is wellknown to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any stateoftheart MIP solver: it needs tailoring to the particular problem; the decomposition must be determined f ..."
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Cited by 9 (6 self)
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DantzigWolfe decomposition is wellknown to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any stateoftheart MIP solver: it needs tailoring to the particular problem; the decomposition must be determined from the typical bordered blockdiagonal matrix structure; the resulting column generation subproblems must be solved efficiently; etc. We provide a computational proofofconcept that the process can be automated in principle, and that strong dual bounds can be obtained on general MIPs for which a solution by a decomposition has not been the first choice. We perform an extensive computational study on the 01 dynamic knapsack problem (without blockdiagonal structure) and on general MIPLIB2003 instances. Our results support that DantzigWolfe reformulation may hold more promise as a generalpurpose tool than previously acknowledged by the research community.
BranchPriceandCut Algorithms
 CONTRIBUTED TO THE WILEY ENCYCLOPEDIA OF OPERATIONS RESEARCH AND MANAGEMENT SCIENCE (EORMS)
, 2010
"... In many mixed integer programs there is some embedded problem structure which can be exploited, often by a decomposition. When the relaxation in each node of a branchandbound tree is solved by column generation, one speaks of branchandprice. Optionally, cutting planes can be added in order to str ..."
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Cited by 7 (0 self)
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In many mixed integer programs there is some embedded problem structure which can be exploited, often by a decomposition. When the relaxation in each node of a branchandbound tree is solved by column generation, one speaks of branchandprice. Optionally, cutting planes can be added in order to strengthen the relaxation, and this is called branchpriceandcut. We introduce the common concepts of convexification and discretization to arrive at a DantzigWolfe type reformulation of a mixed integer program. The relation between the original and the extended formulations helps us understand how cutting planes should be formulated and how branching decisions can be taken while keeping the column generation subproblems manageable.
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
Could we use a million cores to solve an integer program?
 Mathematical Methods of Operations Research
, 2012
"... Abstract Given the steady increase in cores per CPU, it is only a matter of time before supercomputers will have a million or more cores. In this article, we investigate the opportunities and challenges that will arise when trying to utilize this vast computing power to solve a single integer linea ..."
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Abstract Given the steady increase in cores per CPU, it is only a matter of time before supercomputers will have a million or more cores. In this article, we investigate the opportunities and challenges that will arise when trying to utilize this vast computing power to solve a single integer linear optimization problem. We also raise the question of whether best practices in sequential solution of ILPs will be effective in massively parallel environments.
Primal Heuristics for BranchandPrice Algorithms
"... In this paper, we present several primal heuristics which we implemented in the branchandprice solver GCG based on the SCIP framework. This involves new heuristics as well as heuristics from the literature that make use of the reformulation yielded by the DantzigWolfe decomposition. We give shor ..."
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Cited by 2 (1 self)
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In this paper, we present several primal heuristics which we implemented in the branchandprice solver GCG based on the SCIP framework. This involves new heuristics as well as heuristics from the literature that make use of the reformulation yielded by the DantzigWolfe decomposition. We give short descriptions of those heuristics and briefly discuss computational results. Furthermore, we give an outlook on current and further development.
What Could a Million Cores Do To Solve Integer Programs?
, 2012
"... Given the steady increase in cores per CPU, it is only a matter of time until supercomputers will have a million or more cores. In this article, we investigate the opportunities and challenges that will arise when trying to utilize this vast computing power to solve a single integer linear optimizat ..."
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Given the steady increase in cores per CPU, it is only a matter of time until supercomputers will have a million or more cores. In this article, we investigate the opportunities and challenges that will arise when trying to utilize this vast computing power to solve a single integer linear optimization problem. We also raise the question of whether best practices in sequential solution of ILPs will be effective in massively parallel environments.
Generic DantzigWolfe Reformulation of Mixed Integer Programs
, 2011
"... DantzigWolfe decomposition (or reformulation) is wellknown to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any stateoftheart MIP solver as it is considered to require structural problem knowledge and tai ..."
Abstract
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DantzigWolfe decomposition (or reformulation) is wellknown to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any stateoftheart MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proofofconcept that the process can be automated. In particular the detection (better: the construction) of a matrix structure that is useful for DantzigWolfe reformulation of a MIP can be accomplished by suitably permuting rows and columns. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the reformulated problem exploiting column generation. Our results support that DantzigWolfe reformulation may hold more promise as a generalpurpose tool than previously acknowledged by the research community.
Computational Experience . . . . Automatic Decomposition in Discrete Optimization
, 2012
"... Branchandprice algorithms based on DantzigWolfe decomposition have shown great success in solving mixed integer linear optimization problems (MILPs) with specific identifiable structure, such as vehicle routing and crew scheduling problems. For unstructured MILPs, the most frequently used methodo ..."
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Branchandprice algorithms based on DantzigWolfe decomposition have shown great success in solving mixed integer linear optimization problems (MILPs) with specific identifiable structure, such as vehicle routing and crew scheduling problems. For unstructured MILPs, the most frequently used methodology is branchandcut, which depends on generation of “generic” classes of valid inequalities to strengthen bounds. There has been little investigation into the development of a similar “generic“ version of branchandprice, though this is possible in principle. One of the most important elements required for such a generic branchandprice algorithm is an automatic method of decomposition. In this paper, we experiment with hypergraph partitioning as a means of performing such automatic decomposition. Computational results explore the potential for applying branchandprice algorithms within generic solvers and provide insight into how to measure the quality of the decomposition and improve it.
Automatic Decomposition and BranchandPrice  A Status Report
"... We provide an overview of our recent efforts to automatize DantzigWolfe reformulation and column generation/branchandprice for structured, largescale integer programs. We present the need for and the benefits from a generic implementation which does not need any user input or expert knowledge. A ..."
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We provide an overview of our recent efforts to automatize DantzigWolfe reformulation and column generation/branchandprice for structured, largescale integer programs. We present the need for and the benefits from a generic implementation which does not need any user input or expert knowledge. A focus is on detecting structures in integer programs which are amenable to a DantzigWolfe reformulation. We give computational results and discuss future research topics.