FULL LENGTH PAPER

Mathematical Programming

, Volume 113, Issue 2, pp 299-344

Comparison of bundle and classical column generation

  • O. BriantAffiliated withGilco
  • , C. LemaréchalAffiliated withINRIA Email author 
  • , Ph. MeurdesoifAffiliated withUniversity of Bordeaux 1; MAB
  • , S. MichelAffiliated withUniversity of Bordeaux 1; MAB
  • , N. PerrotAffiliated withUniversity of Bordeaux 1; MAB
  • , F. VanderbeckAffiliated withUniversity of Bordeaux 1; MAB

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Abstract

When a column generation approach is applied to decomposable mixed integer programming problems, it is standard to formulate and solve the master problem as a linear program. Seen in the dual space, this results in the algorithm known in the nonlinear programming community as the cutting-plane algorithm of Kelley and Cheney-Goldstein. However, more stable methods with better theoretical convergence rates are known and have been used as alternatives to this standard. One of them is the bundle method; our aim is to illustrate its differences with Kelley’s method. In the process we review alternative stabilization techniques used in column generation, comparing them from both primal and dual points of view. Numerical comparisons are presented for five applications: cutting stock (which includes bin packing), vertex coloring, capacitated vehicle routing, multi-item lot sizing, and traveling salesman. We also give a sketchy comparison with the volume algorithm.

Keywords

Lagrangian duality Dantzig–Wolfe decomposition Stabilized column generation Cutting-plane algorithms Bundle algorithm Volume algorithm Nonsmooth convex optimization

Mathematics Subject Classification (2000)

66K05 90C25 90C27