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Preprint 48-2003

Combinatorial Optimization & Graph Algorithms group (COGA-Preprints)

A Primer in Column Generation
primary: 90C10 Integer programming
secondary: 90-02 Research exposition
90C05 Linear programming
90C06 Large-scale problems
90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut
49M27 Decomposition methods
65K05 Mathematical programming algorithms
Linear Programming, Large Scale Problems, Integer Programming, Branch-and-Bound, Decomposition Methods
We give a didactic introduction to the use of the column generation technique in linear and in particular in integer programming. We touch on both, the relevant basic theory and more advanced ideas which help in solving large scale practical problems. Our discussion includes embedding Dantzig-Wolfe decomposition and Lagrangian relaxation within a branch-and-bound framework, deriving natural branching and cutting rules by means of a so-called compact formulation, and understanding and influencing the behavior of the dual variables during column generation. Most concepts are illustrated via a small example. We close with a discussion of the classical cutting stock problem and some suggestions for further reading.
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