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

## 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.

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