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


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.


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 



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.


  1. 1.
    Altman, A., Kiwiel, K.C.: A note on some analytic center cutting plane methods for convex feasibility and minimization problems. Comput. Optim. Appl. 5(2), 175–180 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alvelos, F., Valério de Carvalho, J.M.: An extended model and a column generation algorithm for the planar multicommodity flow problem. Networks 50(1), 3–16 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Ariyawansa, K., Felt, A.J.: On a new collection of stochastic linear programming test problems. INFORMS J. Comput. 16(3), 291–299 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Babonneau, F., Beltran, C., Haurie, A., Tadonki, C., Vial, J.P.: Proximal-ACCPM: a versatile oracle based optimisation method. In: Kontoghiorghes, E.J., Gatu, C., Amman, H., Rustem, B., Deissenberg, C., Farley, A., Gilli, M., Kendrick, D., Luenberger, D., Maes, R., Maros, I., Mulvey, J., Nagurney, A., Nielsen, S., Pau, L., Tse, E., Whinston, A. (eds.) Optimisation, Econometric and Financial Analysis, Advances in Computational Management Science, vol. 9, pp. 67–89. Springer, Berlin (2007)CrossRefGoogle Scholar
  5. 5.
    Babonneau, F., du Merle, O., Vial, J.P.: Solving large-scale linear multicommodity flow problems with an active set strategy and proximal-ACCPM. Oper. Res. 54(1), 184–197 (2006)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Babonneau, F., Vial, J.P.: ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems. Math. Program. 120, 179–210 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Bach, F.R., Lanckriet, G.R.G., Jordan, M.I.: Multiple kernel learning, conic duality, and the SMO algorithm. In: Proceedings of the twenty-first international conference on Machine learning, ICML ’04, p. 6. ACM, New York (2004)Google Scholar
  8. 8.
    Bahn, O., Merle, O., Goffin, J.L., Vial, J.P.: A cutting plane method from analytic centers for stochastic programming. Math. Program. 69, 45–73 (1995)zbMATHGoogle Scholar
  9. 9.
    Ben-Hur, A., Weston, J.: A user’s guide to support vector machines. In: Data Mining Techniques for the Life Sciences, pp. 223–239. Springer, Berlin (2010)Google Scholar
  10. 10.
    Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4, 238–252 (1962)CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    Birge, J.R., Dempster, M.A., Gassmann, H.I., Gunn, E.A., King, A.J., Wallace, S.W.: A standard input format for multiperiod stochastic linear programs. COAL Newsl. 17, 1–19 (1987)Google Scholar
  12. 12.
    Birge, J.R., Louveaux, F.V.: A multicut algorithm for two-stage stochastic linear programs. Eur. J. Oper. Res. 34(3), 384–392 (1988)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Birge, J.R., Louveaux, F.V.: Introduction to Stochastic Programming. Springer, Berlin (1997)zbMATHGoogle Scholar
  14. 14.
    Briant, O., Lemaréchal, C., Meurdesoif, P., Michel, S., Perrot, N., Vanderbeck, F.: Comparison of bundle and classical column generation. Math. Program. 113, 299–344 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Castro, J.: Solving difficult multicommodity problems with a specialized interior-point algorithm. Ann. Oper. Res. 124, 35–48 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Castro, J., Cuesta, J.: Improving an interior-point algorithm for multicommodity flows by quadratic regularizations. Networks 59(1), 117–131 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Dantzig, G.B.: Linear Programming and its Extensions. Princeton University Press, Princeton (1963)Google Scholar
  18. 18.
    Dantzig, G.B., Madansky, A.: On the solution of two-stage linear programs under uncertainty. In: Proceedings Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 165–176. University of California Press, Berkeley (1961)Google Scholar
  19. 19.
    Dantzig, G.B., Wolfe, P.: The decomposition algorithm for linear programs. Econometrica 29(4), 767–778 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Ellison, E., Mitra, G., Zverovich, V.: FortSP: A Stochastic Programming Solver. OptiRisk Systems, UK (2010)Google Scholar
  22. 22.
    Ford, L.R., Fulkerson, D.R.: A suggested computation for maximal multi-commodity network flows. Manag. Sci. 5(1), 97–101 (1958)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Frangioni, A.: Generalized bundle methods. SIAM J. Optim. 13, 117–156 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Frangioni, A., Gallo, G.: A bundle type dual-ascent approach to linear multicommodity min-cost flow problems. INFORMS J. Comput. 11(4), 370–393 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Frangioni, A., Gendron, B.: A stabilized structured Dantzig–Wolfe decomposition method. Math. Program. 140(1), 45–76 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Frank, A., Asuncion, A.: UCI machine learning repository (2010).
  27. 27.
    Geoffrion, A.M.: Elements of large-scale mathematical programming Part I: concepts. Manag. Sci. 16(11), 652–675 (1970)CrossRefMathSciNetzbMATHGoogle Scholar
  28. 28.
    Geoffrion, A.M.: Elements of large-scale mathematical programming Part II: synthesis of algorithms and bibliography. Manag. Sci. 16(11), 676–691 (1970)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting-stock problem. Oper. Res. 9(6), 849–859 (1961)CrossRefMathSciNetzbMATHGoogle Scholar
  30. 30.
    Goffin, J.L., Gondzio, J., Sarkissian, R., Vial, J.P.: Solving nonlinear multicommodity flow problems by the analytic center cutting plane method. Math. Program. 76, 131–154 (1996)MathSciNetGoogle Scholar
  31. 31.
    Goffin, J.L., Haurie, A., Vial, J.P.: Decomposition and nondifferentiable optimization with the projective algorithm. Manag. Sci. 38(2), 284–302 (1992)CrossRefzbMATHGoogle Scholar
  32. 32.
    Goffin, J.L., Luo, Z.Q., Ye, Y.: Complexity analysis of an interior cutting plane method for convex feasibility problems. SIAM J. Optim. 6(3), 638–652 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Goffin, J.L., Vial, J.P.: Convex nondifferentiable optimization: a survey focused on the analytic center cutting plane method. Optim. Methods Softw. 17, 805–868 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  34. 34.
    Gondzio, J.: Warm start of the primal-dual method applied in the cutting-plane scheme. Math. Program. 83, 125–143 (1998)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Gondzio, J.: Interior point methods 25 years later. Eur. J. Oper. Res. 218(3), 587–601 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Gondzio, J., González-Brevis, P.: A new warmstarting strategy for the primal-dual column generation method. Math. Program. 152(1–2), 113–146 (2015). doi: 10.1007/s10107-014-0779-8
  37. 37.
    Gondzio, J., González-Brevis, P., Munari, P.: New developments in the primal-dual column generation technique. Eur. J. Oper. Res. 224(1), 41–51 (2013)CrossRefzbMATHGoogle Scholar
  38. 38.
    Gondzio, J., Sarkissian, R.: Column generation with a primal-dual method. Technical Report 96.6, Logilab (1996)Google Scholar
  39. 39.
    Gönen, M., Alpaydin, E.: Multiple kernel learning algorithms. J. Mach. Learn. Res. 12, 2211–2268 (2011)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms II: Advanced Theory and Bundle Methods. Springer, Berlin (1993)zbMATHGoogle Scholar
  41. 41.
    Holmes, D.: A (PO)rtable (S)tochastic programming (T)est (S)et (POSTS) (1995). Available in: Accessed April 2013
  42. 42.
    Kall, P., Wallace, S.W.: Stochastic Programming. Wiley, New York (1994)zbMATHGoogle Scholar
  43. 43.
    Kelley, L.E.: The cutting-plane method for solving convex programs. J. Soc. Ind. Appl. Math. 8(4), 703–712 (1960)CrossRefMathSciNetGoogle Scholar
  44. 44.
    Kiwiel, K.C.: Proximity control in bundle methods for convex nondifferentiable minimization. Math. Program. 46, 105–122 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  45. 45.
    Kiwiel, K.C.: Complexity of some cutting plane methods that use analytic centers. Math. Program. 74(1), 47–54 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  46. 46.
    Lanckriet, G., Cristianini, N., Bartlett, P., Ghaoui, L., Jordan, M.: Learning the kernel matrix with semidefinite programming. J. Mach. Learn. Res. 5, 27–72 (2004)zbMATHGoogle Scholar
  47. 47.
    Larsson, T., Yuan, D.: An augmented lagrangian algorithm for large scale multicommodity routing. Comput. Optim. Appl. 27, 187–215 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  48. 48.
    Lemaréchal, C., Nemirovskii, A., Nesterov, Y.: New variants of bundle methods. Math. Program. 69(1–3), 111–147 (1995)CrossRefzbMATHGoogle Scholar
  49. 49.
    Lemaréchal, C., Ouorou, A., Petrou, G.: A bundle-type algorithm for routing in telecommunication data networks. Comput. Optim. Appl. 44, 385–409 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  50. 50.
    Lobo, M.S., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284(1–3), 193–228 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  51. 51.
    Lübbecke, M.E., Desrosiers, J.: Selected topics in column generation. Oper. Res. 53(6), 1007–1023 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  52. 52.
    Marsten, R.E., Hogan, W.W., Blankenship, J.W.: The boxstep method for large-scale optimization. Oper. Res. 23(3), 389–405 (1975)CrossRefMathSciNetzbMATHGoogle Scholar
  53. 53.
    Martinson, R.K., Tind, J.: An interior point method in Dantzig–Wolfe decomposition. Comput. Oper. Res. 26, 1195–1216 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  54. 54.
    McBride, R.D.: Progress made in solving the multicommodity flow problem. SIAM J. Optim. 8(4), 947–955 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  55. 55.
    du Merle, O., Villeneuve, D., Desrosiers, J., Hansen, P.: Stabilized column generation. Discrete Math. 194(1–3), 229–237 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  56. 56.
    Mitchell, J.E., Borchers, B.: Solving real-world linear ordering problems using a primal-dual interior point cutting plane method. Ann. Oper. Res. 62, 253–276 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  57. 57.
    Munari, P., Gondzio, J.: Using the primal-dual interior point algorithm within the branch-price-and-cut method. Comput. Oper. Res. 40(8), 2026–2036 (2013)CrossRefMathSciNetGoogle Scholar
  58. 58.
    Neame, P.: Nonsmooth dual methods in integer programming. Ph.D. thesis, University of Melbourne, Department of Mathematics and Statistics (2000)Google Scholar
  59. 59.
    Ouorou, A., Mahey, P., Vial, J.P.: A survey of algorithms for convex multicommodity flow problems. Manag. Sci. 46(1), 126–147 (2000)CrossRefzbMATHGoogle Scholar
  60. 60.
    Rakotomamonjy, A., Bach, F., Canu, S., Grandvalet, Y.: SimpleMKL. J. Mach. Learn. Res. 9, 2491–2521 (2008)MathSciNetzbMATHGoogle Scholar
  61. 61.
    Ruszczyński, A.: A regularized decomposition method for minimizing a sum of polyhedral functions. Math. Program. 35, 309–333 (1986)CrossRefzbMATHGoogle Scholar
  62. 62.
    Schramm, H., Zowe, J.: A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results. SIAM J. Optim. 2(1), 121–152 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  63. 63.
    Sonnenburg, S., Rätsch, G., Henschel, S., Widmer, C., Behr, J., Zien, A., de Bona, F., Binder, A., Gehl, C., Franc, V.: The SHOGUN machine learning toolbox. J. Mach. Learn. Res. 11, 1799–1802 (2010)zbMATHGoogle Scholar
  64. 64.
    Sonnenburg, S., Rätsch, G., Schäfer, C., Schölkopf, B.: Large scale multiple kernel learning. J. Mach. Learn. Res. 7, 1531–1565 (2006)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Suzuki, T., Tomioka, R.: SpicyMKL: a fast algorithm for multiple kernel learning with thousands of kernels. Mach. Learn. 85, 77–108 (2011)CrossRefMathSciNetzbMATHGoogle Scholar
  66. 66.
    Van Slyke, R., Wets, R.: L-shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17(4), 638–663 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  67. 67.
    Vanderbeck, F.: Implementing mixed integer column generation. In: Desaulniers, G., Desrosiers, J., Solomon, M.M. (eds.) Column Generation, pp. 331–358. Springer, USA (2005)CrossRefGoogle Scholar
  68. 68.
    Vapnik, V.: Statistical Learning Theory. Wiley, New York (1998)zbMATHGoogle Scholar
  69. 69.
    Wentges, P.: Weighted Dantzig–Wolfe decomposition for linear mixed-integer programming. Int. Trans. Oper. Res. 4(2), 151–162 (1997)zbMATHGoogle Scholar
  70. 70.
    Xu, Z., Jin, R., King, I., Lyu, M.: An extended level method for efficient multiple kernel learning. Adv. Neural Inf. Process. Syst. 21, 1825–1832 (2009)Google Scholar
  71. 71.
    Zien, A., Ong, C.S.: Multiclass multiple kernel learning. In: Proceedings of the 24th International Conference on Machine Learning. ICML ’07, pp. 1191–1198. ACM, New York (2007)Google Scholar
  72. 72.
    Zverovich, V., Fábián, C.I., Ellison, E.F., Mitra, G.: A computational study of a solver system for processing two-stage stochastic LPs with enhanced Benders decomposition. Math. Program. Comput. 4, 211–238 (2012)CrossRefMathSciNetzbMATHGoogle Scholar

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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