Mathematical Programming

, Volume 157, Issue 1, pp 69–93 | Cite as

Minimum cardinality non-anticipativity constraint sets for multistage stochastic programming

  • Natashia Boland
  • Irina Dumitrescu
  • Gary Froyland
  • Thomas Kalinowski
Full Length Paper Series B


We consider multistage stochastic programs, in which decisions can adapt over time, (i.e., at each stage), in response to observation of one or more random variables (uncertain parameters). The case that the time at which each observation occurs is decision-dependent, known as stochastic programming with endogeneous observation of uncertainty, presents particular challenges in handling non-anticipativity. Although such stochastic programs can be tackled by using binary variables to model the time at which each endogenous uncertain parameter is observed, the consequent conditional non-anticipativity constraints form a very large class, with cardinality in the order of the square of the number of scenarios. However, depending on the properties of the set of scenarios considered, only very few of these constraints may be required for validity of the model. Here we characterize minimal sufficient sets of non-anticipativity constraints, and prove that their matroid structure enables sets of minimum cardinality to be found efficiently, under general conditions on the structure of the scenario set.


Stochastic programming Endogeneous uncertainty Multistage stochastic programming 



The authors are very grateful to BHP Billiton Ltd. and in particular to Merab Menabde, Peter Stone, and Mark Zuckerberg for their support of the mining-related research that inspired this work. We also thank Hamish Waterer and Laana Giles for useful discussions in the early stages of the research, and thank Hamish for his helpful suggestions and proof-reading of early versions of this paper. We are most grateful to Ignacio Grossmann for his advice and encouragement in completing the paper. This research would not have been possible without the support of the Australian Research Council, grant LP0561744. Finally, the efforts of two anonymous reviewers in improving the paper were greatly appreciated.


  1. 1.
    Artstein, Z., Wets, R.J.-B.: Sensors and information in optimization under stochastic uncertainty. Math. Oper. Res. 28, 523–547 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Asamov, T., Ruszczynński, A.: Time-consistent approximations of risk-averse multistage stochastic optimization problems. Math. Progr. 1–35 (2014). doi: 10.1007/s10107-014-0813-x
  3. 3.
    Bertsimas, D., Georghiou, A.: Design of near optimal decision rules in multistage adaptive mixed-integer optimization. Oper. Res. 1–18 (2015). doi: 10.1287/opre.2015.1365
  4. 4.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, Berlin (1997)zbMATHGoogle Scholar
  5. 5.
    Boland, N., Dumitrescu, I., Froyland, G.: A multistage stochastic programming approach to open pit mine production scheduling with uncertain geology. Optim. Online (2008).
  6. 6.
    Bruni, M.E., Beraldi, P., Conforti, D.: A stochastic programming approach for operating theatre scheduling under uncertainty. IMA J. Manag. Math. 26(1), 99–119 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Colvin, M., Maravelias, C.T.: A stochastic programming approach for clinical trial planning in new drug development. Comput. Chem. Eng. 32, 2626–2642 (2008)CrossRefGoogle Scholar
  8. 8.
    Colvin, M., Maravelias, C.T.: Scheduling of testing tasks and resource planning in new product development using stochastic programming. Comput. Chem. Eng. 33(5), 964–976 (2009)CrossRefGoogle Scholar
  9. 9.
    Colvin, M., Maravelias, C.T.: Modeling methods and a branch and cut algorithm for pharmaceutical clinical trial planning using stochastic programming. Eur. J. Oper. Res. 203, 205–215 (2010)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fragnière, E., Gondzio, J., Yang, X.: Operations risk management by optimally planning the qualified workforce capacity. Eur. J. Oper. Res. 202, 518–527 (2010)CrossRefzbMATHGoogle Scholar
  11. 11.
    Georghiou, A., Wiesemann, W., Kuhn, D.: Generalized decision rule approximations for stochastic programming via liftings. Math. Progr. 1–38 (2014). doi: 10.1007/s10107-014-0789-6
  12. 12.
    Giles, L.: A Multi-stage Stochastic Model for Hydrogeological Optimisation, Masters Thesis, Department of Engineering Science, The University of Auckland (2009)Google Scholar
  13. 13.
    Goel, V., Grossmann, I.E.: A stochastic programming approach to planning of offshore gas field developments under uncertainty in reserves. Comput. Chem. Eng. 28(8), 1409–1429 (2004)CrossRefGoogle Scholar
  14. 14.
    Goel, V., Grossmann, I.E.: A class of stochastic programs with decision dependent uncertainty. Math. Progr. Ser. B 108, 355–394 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Goel, V., Grossmann, I.E., El-Bakry, A.S., Mulkay, E.L.: A novel branch and bound algorithm for optimal development of gas fields under uncertainty in reserves. Comput. Chem. Eng. 30, 1076–1092 (2006)CrossRefGoogle Scholar
  16. 16.
    Guigues, V., Sagastizábal, C.: Risk-averse feasible policies for large-scale multistage stochastic linear programs. Math. Program. 138, 167–198 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gupta, V.: Modeling and Computational Strategies for Optimal Oilfield Development Planning Under Fiscal Rules and Endogenous Uncertainties, PhD Thesis, Carnegie Mellon University (2013)Google Scholar
  18. 18.
    Gupta, V., Grossmann, I.E.: Solution strategies for multistage stochastic programming with endogenous uncertainties. Comput. Chem. Eng. 35, 2235–2247 (2011)CrossRefGoogle Scholar
  19. 19.
    Gupta, V., Grossmann, I.E.: A new decomposition algorithm for multistage stochastic programs with endogenous uncertainties. Comput. Chem. Eng. 62, 62–79 (2014)CrossRefGoogle Scholar
  20. 20.
    Higle, J.L., Rayco, B., Sen, S.: Stochastic scenario decomposition for multistage stochastic programs. IMA J. Manag. Math. 21(1), 39–66 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Jonsbråten, T.W., Wets, R.J.-B., Woodruff, D.L.: A class of stochastic programs with decision dependent random elements. Ann. Oper. Res. 82, 83–106 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kozmík, V., Morton, D.P.: Evaluating policies in risk-averse stochastic dual dynamic programming. Math. Progr. 1–26 (2014). doi: 10.1007/s10107-014-0787-8
  23. 23.
    Philpott, A.B., de Matos, V.L.: Dynamic sampling algorithms for multi-stage stochastic programs with risk aversion. Eur. J. Oper. Res. 218, 470–483 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ramazan, S., Dimitrakopoulos, R.: Production scheduling with uncertain supply: a new solution to the open pit mining problem. Optim. Eng 14, 361–380 (2013)CrossRefzbMATHGoogle Scholar
  25. 25.
    Ruszczyński, A.: Decomposition methods. In: Ruszczyński, A., Shapiro, A. (eds.) Stochastic Programming, Handbook in OR & MS, vol. 10. North-Holland Publishing Company, Amsterdam (2003)Google Scholar
  26. 26.
    Sahinidis, N.V.: Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng. 28(6–7), 971–983 (2004)CrossRefGoogle Scholar
  27. 27.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Volume B. Springer, Berlin (2003)zbMATHGoogle Scholar
  28. 28.
    Schultz, R.: Stochastic programming with integer variables. Math. Program. 97(12), 285–309 (2003)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sen, S., Zhou, Z.: Multistage stochastic decomposition: a bridge between stochastic programming and approximate dynamic programming. SIAM J. Optim. 24(1), 127–153 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Shapiro, A.: On complexity of multistage stochastic programs. Oper. Res. Lett. 34, 1–8 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Shapiro, A.: Analysis of stochastic dual dynamic programming method. Eur. J. Oper. Res. 209(1), 63–72 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Shapiro, A., Dentcheva, D., Ruszczyski, A.P.: Lectures on Stochastic Programming: Modeling and Theory, Vol. 9. SIAM (2009)Google Scholar
  33. 33.
    Shapiro, A., Tekaya, W., da Costa, J.P., Soares, M.P.: Risk neutral and risk averse stochastic dual dynamic programming method. Eur. J. Oper. Res. 224(2), 375–391 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Solak, S., Clarke, J.-P.B., Johnson, E.L., Barnes, E.R.: Optimization of R&D project portfolios under endogenous uncertainty. Eur. J. Oper. Res. 207, 420–433 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Tarhan, B., Grossmann, I.E.: A multistage stochastic programming approach with strategies for uncertainty reduction in the synthesis of process networks with uncertain yields. Comput. Chem. Eng. 32, 766–788 (2008)CrossRefGoogle Scholar
  36. 36.
    Tarhan, B., Grossmann, I.E., Goel, V.: Stochastic programming approach for the planning of offshore oil or gas field infrastructure under decision-dependent uncertainty. Ind. Eng. Chem. Res. 48(6), 3078–3097 (2009)CrossRefGoogle Scholar
  37. 37.
    Tarhan, B., Grossmann, I.E., Goel, V.: Computational strategies for non-convex multistage MINLP models with decision-dependent uncertainty and gradual uncertainty resolution. Ann. Oper. Res. 203(1), 141–166 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Vayanos, P., Kuhn, D., Rustem, B.: Decision rules for information discovery in multi-stage stochastic programming. In: Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC–ECC), Orlando, FL, December 12-15, 2011, pp. 7368–7373Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • Natashia Boland
    • 1
    • 6
  • Irina Dumitrescu
    • 2
    • 3
  • Gary Froyland
    • 4
  • Thomas Kalinowski
    • 5
  1. 1.The University of NewcastleCallaghanAustralia
  2. 2.IBM Research - AustraliaCarltonAustralia
  3. 3.University of MelbourneMelbourneAustralia
  4. 4.The University of New South WalesSydneyAustralia
  5. 5.The University of NewcastleCallaghanAustralia
  6. 6.H. Milton Stewart School of Industrial & Systems Engineering Georgia Institute of TechnologyAtlantaUSA

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