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Mathematical Programming

, Volume 163, Issue 1–2, pp 411–444 | Cite as

A cutting-plane approach for large-scale capacitated multi-period facility location using a specialized interior-point method

  • Jordi Castro
  • Stefano Nasini
  • Francisco Saldanha-da-Gama
Full Length Paper Series A

Abstract

We propose a cutting-plane approach (namely, Benders decomposition) for a class of capacitated multi-period facility location problems. The novelty of this approach lies on the use of a specialized interior-point method for solving the Benders subproblems. The primal block-angular structure of the resulting linear optimization problems is exploited by the interior-point method, allowing the (either exact or inexact) efficient solution of large instances. The consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically, providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting-plane method. The methodology proposed allowed the solution of instances of up to 200 potential locations, one million customers and three periods, resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables. Those problems were solved by the specialized approach in less than one hour and a half, outperforming other state-of-the-art methods, which exhausted the (144 GB of) available memory in the largest instances.

Keywords

Mixed integer linear optimization Interior-point methods Multi-period facility location Cutting planes Benders decomposition Large-scale optimization 

Mathematics Subject Classification

90C06 90C11 90C51 90B80 

Notes

Acknowledgments

The first author has been supported by MINECO/FEDER Grants MTM2012-31440 and MTM2015-65362-R of the Spanish Ministry of Economy and Competitiveness; the second author has been supported by the European Research Council-ref. ERC-2011-StG 283300-REACTOPS; the third author has been supported by the Portuguese Science Foundation (FCT-Fundação para a Ciência e Tecnologia) under the Project UID/MAT/04561/2013 (CMAF-CIO/FCUL). The authors would like to thank the two anonymous reviewers for their valuable comments, suggestions and insights that helped improving the manuscript.

References

  1. 1.
    Albareda-Sambola, M., Fernández, E., Hinojosa, Y., Puerto, J.: The multi-period incremental service facility location problem. Comput. Oper. Res. 36, 1356–1375 (2009)CrossRefzbMATHGoogle Scholar
  2. 2.
    Albareda-Sambola, M., Alonso-Ayuso, A., Escudero, L., Fernández, E., Hinojosa, Y., Pizarro-Romero, C.: A computational comparison of several formulations for the multi-period incremental service facility location problem. TOP 18, 62–80 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alumur, S., Kara, B.Y., Melo, T.: Location and logistics. In: Laporte, G., Nickel, S., Saldanha-da-Gama, F. (eds.) Location Science. Springer, Berlin (2015)Google Scholar
  4. 4.
    Barahona, F., Anbil, R.: The volume algorithm: producing primal solutions with a subgradient method. Math. Program. 87, 385–399 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Comput. Manag. Sci. 2, 3–19 (2005). (English translation of the original paper appeared in. Numerische Mathematik 4(238–252), 1962)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cao, Y., Laird, C.D., Zavala, V.M.: Clustering-based preconditioning for stochastic programs. Comput. Optim. Appl. 64, 379–406 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Castro, J.: A specialized interior-point algorithm for multicommodity network flows. SIAM J. Optim. 10, 852–877 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Castro, J.: An interior-point approach for primal block-angular problems. Comput. Optim. Appl. 36, 195–219 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Castro, J.: Interior-point solver for convex separable block-angular problems. Optim. Methods Softw. 31, 88–109 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Castro, J., Cuesta, J.: Quadratic regularizations in an interior-point method for primal block-angular problems. Math. Program. A 130, 415–445 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Castro, J., Nasini, S.: Mathematical programming approaches for classes of random network problems. Eur. J. Oper. Res. 245, 402–414 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chvátal, V.: Linear Programming. W.H. Freeman and Company, New York (1983)zbMATHGoogle Scholar
  13. 13.
    Colombo, M., Gondzio, J., Grothey, A.: A warm-start approach for large-scale stochastic linear programs. Math. Program. 127, 371–397 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Cornuéjols, G.P., Nemhauser, G.L., Wolsey, L.A.: The uncapacitated facility location problem. In: Mirchandani, P.B., Francis, R.L. (eds.) Discrete Location Theory. Wiley-Interscience, New York (1990)Google Scholar
  15. 15.
    Escudero, L.F., Garín, M.A., Pérez, G., Unzueta, A.: Lagrangian decomposition for large-scale two-stage stochastic mixed 0–1 problems. TOP 20, 347–374 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Fischetti, M., Ljubić, I., Sinnl, M.: Benders decomposition without separability: a computational study for capacitated facility location problems. Eur. J. Oper. Res. 253, 557–569 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fischetti, M., Ljubić, I., Sinnl, M.: Redesigning Benders decomposition for large scale facility location. Manag. Sci. doi: 10.1287/mnsc.2016.2461 (2016)
  18. 18.
    Gondzio, J.: Multiple centrality corrections in a primal-dual method for linear programming. Comput. Optim. Appl. 6, 137–156 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Gondzio, J., Sarkissian, R.: Parallel interior-point solver for structured linear programs. Math. Program. 96, 561–584 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Gondzio, J., Vial, J.-P.: Warm start and \(\epsilon \)-subgradients in the cutting plane scheme for block-angular linear programs. Comput. Optim. Appl. 14, 17–36 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gondzio, J., González-Brevis, P., Munari, P.: New developments in the primal-dual column generation technique. Eur. J. Oper. Res. 224, 41–51 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Held, M., Wolfe, P., Crowder, H.: Validation of subgradient optimization. Math. Program. 6, 62–88 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jena, S., Cordeau, J.-F., Gendron, B.: Modeling and solving a logging camp location problem. Ann. Oper. Res. 232, 151–177 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Jena, S., Cordeau, J.-F., Gendron, B.: Dynamic facility location with generalized modular capacity. Transp. Sci. 49, 484–499 (2015)CrossRefGoogle Scholar
  25. 25.
    Lemaréchal, C., Strodiot, J.J., Bihain, A.: On a bundle algorithm for nonsmooth optimization. In: Mangasarian, O.L., Meyer, R.R., Robinson, S.M. (eds.) Nonlinear Programming 4, pp. 245–282. Academic Press, New York (1981)CrossRefGoogle Scholar
  26. 26.
    Lübbecke, M.E.: Column generation. In: Cochran, J.J., Cox, L.A., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopedia of Operations Research and Management Science, Wiley Online Library, Chichester (2011)Google Scholar
  27. 27.
    Lubin, M., Hall, J.A., Petra, C.G., Anitescu, M.: Parallel distributed-memory simplex for large-scale stochastic LP problems. Comput. Optim. Appl. 55, 571–596 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Magnanti, T.L., Wong, R.T.: Accelerating Benders decomposition: algorithmic enhancement and model selection criteria. Oper. Res. 29, 464–484 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Magnanti, T.L., Wong, R.T.: Decomposition methods for facility location problems. In: Mirchandani, P.B., Francis, R.L. (eds.) Discrete Location Theory. Wiley, New York (1990)Google Scholar
  30. 30.
    Malick, J., de Oliveira, W., Zaourar, S.: Nonsmooth optimization using uncontrolled inexact information. In: Technical Report, INRIA Grenoble (2013). http://www.optimization-online.org/DBHTML/2013/05/3892.html
  31. 31.
    Medhi, D.: Bundle-based decomposition for large-scale convex optimization: error estimate and application to block-angular linear programs. Math. Program. 66, 79–101 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Melo, M.T., Nickel, S., Saldanha-da-Gama, F.: Dynamic multi-commodity capacitated facility location: a mathematical modeling framework for strategic supply chain planning. Comput. Oper. Res. 33, 181–208 (2006)CrossRefzbMATHGoogle Scholar
  33. 33.
    Melo, M.T., Nickel, S., Saldanha-da-Gama, F.: Facility location and supply chain management—a review. Eur. J. Oper. Res. 196, 401–412 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Munari, P., Gondzio, J.: Using the primal-dual interior point algorithm within the branch-price-and-cut method. Comput. Oper. Res. 40, 2026–2036 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Nickel, S., Saldanha-da-Gama, F.: Multi-period facility location. In: Laporte, G., Nickel, S., Saldanha-da-Gama, F. (eds.) Location Science. Springer, Berlin (2015)Google Scholar
  37. 37.
    Nickel, S., Saldanha-da-Gama, F., Ziegler, H.-P.: A multi-stage stochastic supply network design problem with financial decisions and risk management. Omega 40, 511–524 (2012)CrossRefGoogle Scholar
  38. 38.
    Oliveira, W., Sagastizábal, C., Scheimberg, S.: Inexact bundle methods for two-stage stochastic programming. SIAM J. Optim. 21, 517–544 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Petra, C.G., Schenk, O., Lubin, M., Gaertner, K.: An augmented incomplete factorization approach for computing the Schur complement in stochastic optimization. SIAM J. Sci. Comput. 36, C139–C162 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Polyak, B.T.: Subgradient method: a survey of Soviet research. In: Lemaréchal, C., Mifflin, R. (eds.) Nonsmooth Optimization, pp. 5–28. Pergamon Press, Oxford (1978)Google Scholar
  41. 41.
    Rei, W., Cordeau, J.-F., Gendreau, M., Soriano, P.: Accelerating Benders decomposition by local branching. INFORMS J. Comput. 21, 333–345 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Robinson, S.M.: Bundle-based decomposition: description and preliminary results. In: Prékopa, A., Szelezsfin, J., Strazicky, B. (eds.) System Modelling and Optimization, Lecture Notes in Control and Information Sciences, vol. 84. Springer, Berlin (1986)Google Scholar
  43. 43.
    Ruszczyński, A.: An augmented Lagrangian decomposition method for block diagonal linear programming problems. Oper. Res. Lett. 8, 287–294 (1989)MathSciNetCrossRefGoogle Scholar
  44. 44.
    van Ackooij, W., Frangioni, A., de Oliveira, W.: Inexact stabilized Benders’ decomposition approaches with application to chance-constrained problems with finite support. Comput. Optim. Appl. doi: 10.1007/s10589-016-9851-z (2016)
  45. 45.
    Wentges, P.: Accelerating Benders decomposition for the capacitated facility location problem. Math. Methods Oper. Res. 44, 267–290 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Wright, S.J.: Primal-Dual Interior-Point Methods. SIAM, Philapelphia (1996)zbMATHGoogle Scholar
  47. 47.
    Zakeri, G., Philpott, A.B., Ryan, D.M.: Inexact cuts in Benders decomposition. SIAM J. Optim. 10, 643–657 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  1. 1.Department of Statistics and Operations ResearchUniversitat Politècnica de CatalunyaBarcelonaSpain
  2. 2.Department of MarketingIESEG School of Management (LEM-CNRS 9221)LilleFrance
  3. 3.Department of Statistics and Operations Research/Centro de Matemática, Aplicações Fundamentais e Investigação Operacional, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal

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