# Minimum cardinality non-anticipativity constraint sets for multistage stochastic programming

- 388 Downloads
- 3 Citations

## Abstract

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.

## Keywords

Stochastic programming Endogeneous uncertainty Multistage stochastic programming## Notes

### Acknowledgments

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.

## References

- 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.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.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.Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, Berlin (1997)zbMATHGoogle Scholar
- 5.Boland, N., Dumitrescu, I., Froyland, G.: A multistage stochastic programming approach to open pit mine production scheduling with uncertain geology. Optim. Online (2008). http://www.optimization-online.org/DB_FILE/2008/10/2123.pdf
- 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.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.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.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.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.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.Giles, L.: A Multi-stage Stochastic Model for Hydrogeological Optimisation, Masters Thesis, Department of Engineering Science, The University of Auckland (2009)Google Scholar
- 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.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.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.Guigues, V., Sagastizábal, C.: Risk-averse feasible policies for large-scale multistage stochastic linear programs. Math. Program.
**138**, 167–198 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Gupta, V., Grossmann, I.E.: Solution strategies for multistage stochastic programming with endogenous uncertainties. Comput. Chem. Eng.
**35**, 2235–2247 (2011)CrossRefGoogle Scholar - 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.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.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.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.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.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.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.Sahinidis, N.V.: Optimization under uncertainty: state-of-the-art and opportunities. Comput. Chem. Eng.
**28**(6–7), 971–983 (2004)CrossRefGoogle Scholar - 27.Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Volume B. Springer, Berlin (2003)zbMATHGoogle Scholar
- 28.Schultz, R.: Stochastic programming with integer variables. Math. Program.
**97**(12), 285–309 (2003)MathSciNetzbMATHGoogle Scholar - 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.Shapiro, A.: On complexity of multistage stochastic programs. Oper. Res. Lett.
**34**, 1–8 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 31.Shapiro, A.: Analysis of stochastic dual dynamic programming method. Eur. J. Oper. Res.
**209**(1), 63–72 (2011)MathSciNetCrossRefzbMATHGoogle Scholar - 32.Shapiro, A., Dentcheva, D., Ruszczyski, A.P.: Lectures on Stochastic Programming: Modeling and Theory, Vol. 9. SIAM (2009)Google Scholar
- 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.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.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.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.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.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