Minimum cardinality non-anticipativity constraint sets for multistage stochastic programming

We consider the case of multistage stochastic programming, in which decisions can adapt over time (i.e. at each stage) in response to observation of one or more random variables (uncertain parameters) and the time at which each observation occurs is decision-dependent. This is the difficult case of stochastic programming with endogeneous observation of uncertainty. 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 characterise 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. This research was supported by the Australian Research Council Linkage Project grant LP0561744 and by BHP Billiton Limited. N. Boland The University of Newcastle, University Drive, Callaghan, NSW 2308, Australia Tel.: +61-2-4921-6171 E-mail: [email protected] I. Dumitrescu IBM Research Australia E-mail: [email protected] G. Froyland The University of New South Wales E-mail: [email protected] T. Kalinowski The University of Newcastle E-mail: [email protected]