Optimization Letters

, Volume 2, Issue 3, pp 377–388 | Cite as

Quantitative stability of fully random mixed-integer two-stage stochastic programs

Original Paper

Abstract

Mixed-integer two-stage stochastic programs with fixed recourse matrix, random recourse costs, technology matrix, and right-hand sides are considered. Quantitative continuity properties of its optimal value and solution set are derived when the underlying probability distribution is perturbed with respect to an appropriate probability metric.

Keywords

Stochastic programming Two-stage Mixed-integer Stability Weak convergence Probability metric Discrepancy 

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References

  1. 1.
    Bank B., Guddat J., Klatte D., Kummer B. and Tammer K. (1982). Non-Linear Parametric Optimization. Akademie-Verlag, Berlin Google Scholar
  2. 2.
    Bank B. and Mandel R. (1988). Parametric Integer Optimization. Akademie-Verlag, Berlin MATHGoogle Scholar
  3. 3.
    Billingsley P. (1968). Convergence of Probability Measures. Wiley, New York MATHGoogle Scholar
  4. 4.
    Blair C.E. and Jeroslow R.G. (1977). The value function of a mixed integer program I. Discrete Math. 19: 121–138 CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Blair C.E. and Jeroslow R.G. (1979). The value function of a mixed integer program II. Discrete Math. 25: 7–19 CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Cook W., Gerards A.M.H., Schrijver A. and Tardos É. (1986). Sensitivity theorems in integer linear programming. Mathem. Programm. 34: 251–264 CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Eichhorn A. and Römisch W. (2007). Stochastic integer programming: limit theorems and confidence intervals. Math. Oper. Res. 32: 118–135 CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Engell S., Märkert A., Sand G. and Schultz R. (2004). Aggregated scheduling of a multiproduct batch plant by two-stage stochastic integer programming. Optim. Eng. 5: 335–359 CrossRefMathSciNetGoogle Scholar
  9. 9.
    Heitsch, H., Römisch, W.: A note on scenario reduction for two-stage stochastic programs. Oper. Res. Lett. (2007) (to appear)Google Scholar
  10. 10.
    Henrion, R., Küchler, C., Römisch, W.: Discrepancy distances and scenario reduction in two-stage stochastic integer programming, Preprint, DFG Research Center Matheon “Mathematics for key technologies” (2007)Google Scholar
  11. 11.
    Klein Haneveld W.K. and van der Vlerk M.H. (1999). Stochastic integer programming: general models and algorithms. Ann. Oper. Res. 85: 39–57 CrossRefMathSciNetMATHGoogle Scholar
  12. 12.
    Louveaux F. and Schultz R. (2003). Stochastic integer programming, in Stochastic Programming, Handbooks in Operations Research and Management Science, vol 10. Elsevier, Amsterdam, 213–266 Google Scholar
  13. 13.
    Nowak M.P., Schultz R. and Westphalen M. (2005). A stochastic integer programming model for incorporating day-ahead trading of electricity into hydro-thermal unit commitment. Optim. Eng. 6: 163–176 CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Nürnberg R and Römisch W. (2002). A two-stage planning model for power scheduling in a hydro-thermal system under uncertainty. Optim. Eng. 3: 355–378 CrossRefMathSciNetMATHGoogle Scholar
  15. 15.
    Rachev S.T. (1991). Probability Metrics and the Stability of Stochastic Models. Wiley, Chichester MATHGoogle Scholar
  16. 16.
    Rachev S.T. and Römisch W. (2002). Quantitative stability in stochastic programming: the method of probability metrics. Math. Oper. Res. 27: 792–818 CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Rockafellar R.T. and Wets R.J-B. (1998). Variational Analysis. Springer, Berlin MATHGoogle Scholar
  18. 18.
    Römisch W.: Stability of stochastic programming problems. In: Stochastic Programming, Handbooks in Operations Research and Management Science, vol. 10, pp. 483–554. Elsevier, Amsterdam (2003)Google Scholar
  19. 19.
    Schultz R. (1995). On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Math. Programm. 70: 73–89 MathSciNetGoogle Scholar
  20. 20.
    Schultz R. (1996). Rates of convergence in stochastic programs with complete integer recourse. SIAM J. Optim. 6: 1138–1152 CrossRefMathSciNetMATHGoogle Scholar
  21. 21.
    Schultz R. (2003). Stochastic programming with integer variables. Math. Programm. 97: 285–309 MathSciNetMATHGoogle Scholar
  22. 22.
    Sen S. (2005). Algorithms for stochastic mixed-integer programming models, Chapter 9 in Discrete Optimization. In: Aardal, K., Nemhauser, G.L. and Weissmantel, R. (eds) Handbooks in Operations Research and Management Science, vol. 12., pp 515–558. Elsevier, Amsterdam Google Scholar
  23. 23.
    Walkup D and Wets R.J.-B. (1969). Lifting projections of convex polyhedra. Pacific J. Math. 28: 465–475 MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Institute of MathematicsHumboldt-University BerlinBerlinGermany

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