Stochastic Mixed-Integer Programming

  • Willem K. Klein Haneveld
  • Maarten H. van der Vlerk
  • Ward Romeijnders
Part of the Graduate Texts in Operations Research book series (GRTOPR)


In this chapter we consider a generalization of the recourse model in Chap.  3, obtained by allowing integrality restrictions on some or all of the decision variables. First we give some motivation why such mixed-integer recourse models are useful and interesting. Following the presentation of the general model, we give several examples of applications. Next we discuss mathematical properties of the general model as well as the so-called simple integer recourse model, which is the analogue of the continuous simple recourse model discussed in Sect.  3.3.2. We conclude this chapter with an overview of available algorithms.


  1. 3.
    C.C. Carøe and J. Tind. L-shaped decomposition of two-stage stochastic programs with integer recourse. Math. Program., 83(3):451–464, 1998.CrossRefGoogle Scholar
  2. 4.
    Claus C. Carøe and Rüdiger Schultz. Dual decomposition in stochastic integer programming. Oper. Res. Lett., 24(1–2):37–45, 1999.CrossRefGoogle Scholar
  3. 9.
    R.L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1:132–133, 1972.CrossRefGoogle Scholar
  4. 17.
    W.K. Klein Haneveld, L. Stougie, and M.H. van der Vlerk. On the convex hull of the composition of a separable and a linear function. Discussion Paper 9570, CORE, Louvain-la-Neuve, Belgium, 1995.Google Scholar
  5. 18.
    W.K. Klein Haneveld, L. Stougie, and M.H. van der Vlerk. An algorithm for the construction of convex hulls in simple integer recourse programming. Ann. Oper. Res., 64:67–81, 1996.CrossRefGoogle Scholar
  6. 19.
    W.K. Klein Haneveld, L. Stougie, and M.H. van der Vlerk. Simple integer recourse models: convexity and convex approximations. Math. Program., 108(2–3):435–473, 2006.CrossRefGoogle Scholar
  7. 20.
    W.K. Klein Haneveld and M.H. van der Vlerk. Stochastic integer programming: General models and algorithms. Ann. Oper. Res., 85:39–57, 1999.CrossRefGoogle Scholar
  8. 23.
    G. Laporte and F.V. Louveaux. The integer L-shaped method for stochastic integer programs with complete recourse. Oper. Res. Lett., 13:133–142, 1993.CrossRefGoogle Scholar
  9. 28.
    R.R. Meyer. On the existence of optimal solutions to integer and mixed-integer programming problems. Mathematical Programming, 7:223–235, 1974.CrossRefGoogle Scholar
  10. 29.
    G.L. Nemhauser and L.A. Wolsey. Integer and Combinatorial Optimization. Wiley, New York, 1988.CrossRefGoogle Scholar
  11. 36.
    W. Romeijnders, M.H. van der Vlerk, and W.K. Klein Haneveld. Convex approximations of totally unimodular integer recourse models: A uniform error bound. SIAM Journal on Optimization, 25:130–158, 2015.CrossRefGoogle Scholar
  12. 37.
    W. Romeijnders, M.H. van der Vlerk, and W.K. Klein Haneveld. Total variation bounds on the expectation of periodic functions with applications to recourse approximations. Math. Program., 157:3–46, 2016b.CrossRefGoogle Scholar
  13. 38.
    A. Schrijver. Theory of Linear and Integer Programming. Wiley, Chichester, 1986.Google Scholar
  14. 39.
    R. Schultz. Continuity and stability in two-stage stochastic integer programming. In K. Marti, editor, Lecture Notes in Economics and Mathematical Systems, volume 379, pages 81–92. Springer-Verlag, Berlin, 1992.Google Scholar
  15. 40.
    R. Schultz. Continuity properties of expectation functions in stochastic integer programming. Math. Oper. Res., 18:578–589, 1993.CrossRefGoogle Scholar
  16. 41.
    R. Schultz. On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Math. Program., 70:73–89, 1995.Google Scholar
  17. 42.
    R. Schultz, L. Stougie, and M.H. van der Vlerk. Solving stochastic programs with complete integer recourse: A framework using Gröbner Bases. Math. Program., 83(2):229–252, 1998.Google Scholar
  18. 44.
    L. Stougie and M.H. van der Vlerk. Stochastic integer programming. In M. Dell’Amico, F. Maffioli, and S. Martello, editors, Annotated Bibliographies in Combinatorial Optimization, chapter 9, pages 127–141. Wiley, 1997.Google Scholar
  19. 47.
    M.H. van der Vlerk. Stochastic programming with integer recourse. PhD thesis, University of Groningen, The Netherlands, 1995.Google Scholar
  20. 48.
    M.H. van der Vlerk. Convex approximations for complete integer recourse models. Math. Program., 99(2):297–310, 2004.CrossRefGoogle Scholar
  21. 49.
    R. Van Slyke and R.J-B. Wets. L-shaped linear programs with applications to control and stochastic programming. SIAM Journal on Applied Mathematics, 17:638–663, 1969.CrossRefGoogle Scholar
  22. 52.
    R.M. Wollmer. Two-stage linear programming under uncertainty with 0–1 first stage variables. Math. Program., 19:279–288, 1980.CrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Willem K. Klein Haneveld
    • 1
  • Maarten H. van der Vlerk
    • 1
  • Ward Romeijnders
    • 1
  1. 1.Department of OperationsUniversity of GroningenGroningenThe Netherlands

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