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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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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
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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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.
Computational experience with generic decomposition using the DIP framework
"... Decomposition methods are techniques for exploiting the tractable substructures of an integer program in order to obtain improved solution techniques. In particular, the fundamental idea is to exploit our ability to either optimize over and/or separate from the convex hull of solutions to a given re ..."
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Decomposition methods are techniques for exploiting the tractable substructures of an integer program in order to obtain improved solution techniques. In particular, the fundamental idea is to exploit our ability to either optimize over and/or separate from the convex hull of solutions to a given relaxation in order to derive improved methods of bounding the optimal solution value.
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 ..."
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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.
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.
EXPLOITING STRUCTURE IN INTEGER PROGRAMS
, 2011
"... This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a wellknown optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear eq ..."
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This dissertation argues the case for exploiting certain structures in integer linear programs. Integer linear programming is a wellknown optimisation problem, which seeks the optimum of a linear function of variables, whose values are required to be integral as well as to satisfy certain linear equalities and inequalities. The state of the art in solvers for this problem is the “branch and bound ” approach. The performance of such solvers depends crucially on four types of inbuilt heuristics: primal, improvement, branching, and cutseparation or, more generally, bounding heuristics. Such heuristics in generalpurpose solvers have not, until recently, exploited structure in integer linear programs beyond the recognition of certain types of singlerow constraints. Many alternative approaches to integer linear programming can be cast in the following, novel framework. “Structure” in any integer linear program
Primal Heuristics for Mixed Integer Programs with a Staircase Structure
"... Abstract. In the recent years, Mixed Integer Programms (MIPs) have become a powerful tool to solve problems from a wide range of applications. When solving largescale MIPs, primal heuristics have gained increasing attention: Ideally, a heuristic finds a good feasible solution in a short amount of ..."
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Abstract. In the recent years, Mixed Integer Programms (MIPs) have become a powerful tool to solve problems from a wide range of applications. When solving largescale MIPs, primal heuristics have gained increasing attention: Ideally, a heuristic finds a good feasible solution in a short amount of time. In this paper, we present three heuristic approaches for MIPs whose coefficient matrices have a socalled staircase structure. We will present and evaluate a generalization of a the rolling horizon heuristic that has recently been successful applied to several structured MIPs as well as a new approach called fifty/fifty. Furthermore, we will investigate whether socalled diving heuristics can be extended to staircase structures.
A DECOMPOSITION METHOD FOR LARGE SCALE MILPS, WITH PERFORMANCE GUARANTEES AND A POWER SYSTEM APPLICATION
"... Abstract. Lagrangian duality in mixed integer optimization is a useful framework for problems decomposition and for producing tight lower bounds to the optimal objective, but in contrast to the convex counterpart, it is generally unable to produce optimal solutions directly. In fact, solutions reco ..."
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Abstract. Lagrangian duality in mixed integer optimization is a useful framework for problems decomposition and for producing tight lower bounds to the optimal objective, but in contrast to the convex counterpart, it is generally unable to produce optimal solutions directly. In fact, solutions recovered from the dual may be not only suboptimal, but even infeasible. In this paper we concentrate on large scale mixed–integer programs with a specific structure that is of practical interest, as it appears in a variety of application domains such as power systems or supply chain management. We propose a solution method for these structures, in which the primal problem is modified in a certain way, guaranteeing that the solutions produced by the corresponding dual are feasible for the original unmodified primal problem. The modification is simple to implement and the method is amenable to distributed computations. We also demonstrate that the quality of the solutions recovered using our procedure improves as the problem size increases, making it particularly useful for large scale instances for which commercial solvers are inadequate. We illustrate the efficacy of our method with extensive experimentations on a problem stemming from power systems. 1.