, Volume 20, Issue 2, pp 347–374 | Cite as

Lagrangian Decomposition for large-scale two-stage stochastic mixed 0-1 problems

  • L. F. Escudero
  • M. A. GarínEmail author
  • G. Pérez
  • A. Unzueta
Original Paper


In this paper we study solution methods for solving the dual problem corresponding to the Lagrangian Decomposition of two-stage stochastic mixed 0-1 models. We represent the two-stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangian Decomposition (LD) is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangian Decomposition schemes: the Subgradient Method, the Volume Algorithm, the Progressive Hedging Algorithm, and the Dynamic Constrained Cutting Plane scheme. We test the tightness of the LD bounds in a testbed of medium- and large-scale stochastic instances.


Two-stage stochastic integer programming Lagrangian Decomposition Subgradient Method Volume Algorithm Progressive Hedging Algorithm and Dynamic Constrained Cutting Plane scheme 

Mathematics Subject Classification (2000)

90 90-08 90C06 90C11 90C15 



This research has been partially supported by the projects ECO2008-00777 ECON from the Ministry of Education and Science, Grupo de Investigación IT-347-10 from the Basque Government, grant FPU ECO-2006 from the Ministry of Education and Science, grants RM URJC-CM-2008-CET-3703 and RIESGOS CM from Comunidad de Madrid, and PLANIN MTM2009-14087-C04-01 from Ministry of Science and Innovation, Spain. We would like to express our gratefulness to Prof. Dr. F. Tusell for making easier the access to the Laboratory of Quantitative Economics (Economic Sciences School, UPV/EHU) to perform and check the computing experience.


  1. Alonso-Ayuso A, Escudero LF, Garín A, Ortuño MT, Pérez G (2003a) An approach for strategic supply chain planning based on stochastic 0–1 programming. J Glob Optim 26:97–124 CrossRefGoogle Scholar
  2. Alonso-Ayuso A, Escudero LF, Ortuño MT (2003b) BFC a Branch-and-Fix Coordination algorithmic framework for solving some types of stochastic pure and mixed 0-1 programs. Eur J Oper Res 151:503–519 CrossRefGoogle Scholar
  3. Barahona F, Anbil R (2000) The Volume algorithm: producing primal solutions with a subgradient method. Math Program 87:385–399 CrossRefGoogle Scholar
  4. Bertsekas DP (1982) Constrained optimization and Lagrange multiplier methods. Academic Press, San Diego Google Scholar
  5. Birge JR, Louveaux FV (1997) Introduction to stochastic programming. Springer, Berlin Google Scholar
  6. Carøe CC, Schultz R (1999) Dual decomposition in stochastic integer programming. Oper Res Lett 24:37–45 CrossRefGoogle Scholar
  7. Carøe CC, Tind J (1998) L-shaped decomposition of two-stage stochastic programs with integer recourse. Math Program 83:451–464 CrossRefGoogle Scholar
  8. Engell S, Märkert A, Sand G, Schultz R (2004) Aggregated scheduling of a multiproduct batch plant by two-stage stochastic integer programming. Optim Eng 5:335–359 CrossRefGoogle Scholar
  9. Escudero LF (2009) On a mixture of the fix-and-relax coordination and Lagrangian substitution schemes for multistage stochastic mixed integer programming. Top 17:5–29 CrossRefGoogle Scholar
  10. Escudero LF, Garín A, Merino M, Pérez G (2009) A general algorithm for solving two-stage stochastic mixed 0-1 first stage problems. Comput Oper Res 36:2590–2600 CrossRefGoogle Scholar
  11. Escudero LF, Garín A, Merino M, Pérez G (2010a) On BFC-MSMIP strategies for scenario cluster partitioning, Twin Node Family branching selection and bounding for multistage stochastic mixed integer programming. Comput Oper Res 37:738–753 CrossRefGoogle Scholar
  12. Escudero LF, Garín A, Merino M, Pérez G (2010b) An exact algorithm for solving large-scale two-stage stochastic mixed integer problems: some theoretical and experimental aspects. Eur J Oper Res 204:105–116 CrossRefGoogle Scholar
  13. Geoffrion AM (1974) Lagrangian relaxation for integer programming. Math Program Stud 2:82–114 CrossRefGoogle Scholar
  14. Guignard M (2003) Lagrangian relaxation. Top 11:151–228 CrossRefGoogle Scholar
  15. Guignard M, Kim S (1987) Lagrangian decomposition. A model yielding stronger Lagrangian bounds. Math Program 39:215–228 CrossRefGoogle Scholar
  16. Held M, Karp RM (1971) The traveling salesman problem and minimum spanning trees: part II. Math Program 1:6–25 CrossRefGoogle Scholar
  17. Held M, Wolfe P, Crowder H (1974) Validation of subgradient optimization. Math Program 6:62–88 CrossRefGoogle Scholar
  18. Helmberg C, Kiwiel KC (2002) A spectral bundle method with bounds. Math Program 93:173–194 CrossRefGoogle Scholar
  19. Hemmecke R, Schultz R (2001) Decomposition methods for two-stage stochastic Integer Programs. In: Grötschel M, Krumke SO, Rambau J (eds) Online optimization of large scale systems, pp 601–622. Springer, Berlin Google Scholar
  20. INFORMS. COIN-OR (2008) COmputational INfrastructure for Operations Research.
  21. Jimenez RN, Conejo AJ (1997) Short-term hydro-thermal coordination by Lagrangian relaxation: solution of the dual problem. IEEE Trans Power Syst 14:89–95 Google Scholar
  22. Kiwiel KC (1990) Proximity control in bundle methods for convex nondifferentiable minimization. Math Program 46:15–122 CrossRefGoogle Scholar
  23. Klein Haneveld W, van der Vlerk Kang M (1999) Stochastic integer programming: general models and algorithms. Ann Oper Res 85:39–57 CrossRefGoogle Scholar
  24. Laporte G, Louveaux FV (2002) An integer L-shaped algorithm for the capacitated vehicle routing problem with stochastic demands. Oper Res 50:415–423 CrossRefGoogle Scholar
  25. Li D, Sun X (2006) Nonlinear integer programming. Springer, Berlin Google Scholar
  26. Ntaimo L, Sen S (2005) The million variable ‘march’ for stochastic combinatorial optimization. J Glob Optim 32:385–400 CrossRefGoogle Scholar
  27. Polyak BT (1987) Introduction to optimization software Google Scholar
  28. Rockafellar RT, Wets RJ-B (1991) Scenario and policy aggregation in optimisation under uncertainty. Math Oper Res 16:119–147 CrossRefGoogle Scholar
  29. Schultz R (2003) Stochastic programming with integer variables. Math Program, Ser B 97:285–309 Google Scholar
  30. Sen S, Higle JL (2005) The C3 theorem and a D2 algorithm for large scale stochastic mixed-integer programming: set convexification. Math Program, Ser A 104:1–20 CrossRefGoogle Scholar
  31. Sen S, Sherali HD (2006) Decomposition with branch-and-cut approaches for two-stage stochastic mixed-integer programming. Math Program, Ser A 106:203–223 CrossRefGoogle Scholar
  32. Sherali HD, Smith JC (2009) Two-stage hierarchical multiple risk problems: models and algorithms. Math Program, Ser A 120:403–427 CrossRefGoogle Scholar
  33. Sherali HD, Zhu X (2006) On solving discrete two stage stochastic programs having mixed-integer first and second stage variables. Math Program, Ser A 108:597–611 CrossRefGoogle Scholar
  34. Takriti S, Birge JR (2000) Lagrangian solution techniques and bounds for loosely coupled mixed-integer stochastic programs. Oper Res 48:91–98 CrossRefGoogle Scholar
  35. Till J, Sand G, Urselmann M, Engell S (2007) A hybrid evolutionary algorithm for solving two-stage stochastic integer programs in chemical batch scheduling. Comput Chem Eng 31:630–647 CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2011

Authors and Affiliations

  • L. F. Escudero
    • 1
  • M. A. Garín
    • 2
    Email author
  • G. Pérez
    • 3
  • A. Unzueta
    • 2
  1. 1.Dpto. Estadística e Investigación OperativaUniversidad Rey Juan CarlosMóstolesSpain
  2. 2.Dpto. de Economía Aplicada IIIUniversidad del País VascoBilbaoSpain
  3. 3.Dpto. de Matemática Aplicada, Estadística e Investigación OperativaUniversidad del País VascoLeioaSpain

Personalised recommendations