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Preprint 25-2003

Combinatorial Optimization & Graph Algorithms group (COGA-Preprints)

Title
On Compact Formulations for Integer Programs Solved by Column Generation
Authors
Daniel Villeneuve, Jacques Desrosiers, Marco E. Lübbecke, and Francois Soumis
Publication
Submitted to: Annals of Operations Research
Classification
MSC:
primary: 90C10 Integer programming
secondary: 49M27 Decomposition methods
65K05 Mathematical programming algorithms
90C06 Large-scale problems
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
Keywords
Integer programming, branch and bound, column generation
Abstract
Column generation has become a powerful tool in solving large scale integer programs. It is well known that most of the often reported compatibility issues between pricing subproblem and branching rule disappear when branching decisions are based on imposing constraints on the subproblem's variables. This can be generalized to branching on variables of a so-called compact formulation. We constructively show that such a formulation always exists under mild assumptions. It has a block diagonal structure with identical subproblems, each of which contributes only one column in an integer solution. This construction has an interpretation as reversing a Dantzig-Wolfe decomposition. Our proposal opens the way for the development of branching rules adapted to the subproblem's structure and to the linking constraints.
Source
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