Mathematical Programming

, Volume 113, Issue 2, pp 299–344

Comparison of bundle and classical column generation

  • O. Briant
  • C. Lemaréchal
  • Ph. Meurdesoif
  • S. Michel
  • N. Perrot
  • F. Vanderbeck
FULL LENGTH PAPER

DOI: 10.1007/s10107-006-0079-z

Cite this article as:
Briant, O., Lemaréchal, C., Meurdesoif, P. et al. Math. Program. (2008) 113: 299. doi:10.1007/s10107-006-0079-z

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 dualityDantzig–Wolfe decompositionStabilized column generationCutting-plane algorithmsBundle algorithmVolume algorithmNonsmooth convex optimization

Mathematics Subject Classification (2000)

66K0590C2590C27

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  • O. Briant
    • 1
  • C. Lemaréchal
    • 2
  • Ph. Meurdesoif
    • 3
  • S. Michel
    • 3
  • N. Perrot
    • 3
  • F. Vanderbeck
    • 3
  1. 1.GilcoGrenobleFrance
  2. 2.INRIASaint IsmierFrance
  3. 3.University of Bordeaux 1; MABTalenceFrance