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Mathematical Programming Computation

, Volume 8, Issue 1, pp 47–82 | Cite as

Large-scale optimization with the primal-dual column generation method

  • Jacek Gondzio
  • Pablo González-Brevis
  • Pedro MunariEmail author
Full Length Paper

Abstract

The primal-dual column generation method (PDCGM) is a general-purpose column generation technique that relies on the primal-dual interior point method to solve the restricted master problems. The use of this interior point method variant allows to obtain suboptimal and well-centered dual solutions which naturally stabilizes the column generation process. As recently presented in the literature, reductions in the number of calls to the oracle and in the CPU times are typically observed when compared to the standard column generation, which relies on extreme optimal dual solutions. However, these results are based on relatively small problems obtained from linear relaxations of combinatorial applications. In this paper, we investigate the behaviour of the PDCGM in a broader context, namely when solving large-scale convex optimization problems. We have selected applications that arise in important real-life contexts such as data analysis (multiple kernel learning problem), decision-making under uncertainty (two-stage stochastic programming problems) and telecommunication and transportation networks (multicommodity network flow problem). In the numerical experiments, we use publicly available benchmark instances to compare the performance of the PDCGM against recent results for different methods presented in the literature, which were the best available results to date. The analysis of these results suggests that the PDCGM offers an attractive alternative over specialized methods since it remains competitive in terms of number of iterations and CPU times even for large-scale optimization problems.

Keywords

Column generation Cutting plane method Interior point methods Convex optimization Multiple kernel learning problem Two-stage stochastic programming Multicommodity network flow problem 

Mathematics Subject Classification

90C06 Large-scale problems 49M27 Decomposition methods  90C25 Convex programming 90C51 Interior-point methods 

Notes

Acknowledgments

We would like to express our gratitude to Victor Zverovich for kindly making available to us some of the TSSP instances included in this study. Also, we would like to thank Robert Gower for proofreading an early version of this paper. We are very thankful to the anonymous referees for their careful reading and the important suggestions made, which certainly helped to improve the first draft of this paper. Pablo González-Brevis has been supported by CONICYT, Chile through FONDECYT grant 11140521. Pedro Munari has been supported by FAPESP (São Paulo Research Foundation, Brazil) through grants 14/00939-8 and 14/50228-0.

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Copyright information

© Springer-Verlag Berlin Heidelberg and The Mathematical Programming Society 2015

Authors and Affiliations

  • Jacek Gondzio
    • 1
  • Pablo González-Brevis
    • 2
  • Pedro Munari
    • 3
    Email author
  1. 1.School of MathematicsThe University of EdinburghEdinburghUK
  2. 2.School of EngineeringUniversidad del DesarrolloConcepciónChile
  3. 3.Production Engineering DepartmentFederal University of São CarlosSão CarlosBrazil

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