Revisiting some results on the complexity of multistage stochastic programs and some extensions

In this work we present explicit definitions for the sample complexity associated with the Sample Average Approximation (SAA) Method for instances and classes of multistage stochastic optimization problems. For such, we follow the same notion firstly considered in Kleywegt et al. (2001). We define the complexity for an arbitrary class of problems by considering its worst case behavior, as it is a common approach in the complexity theory literature. We review some sample complexity results for the SAA method obtained so far in the literature, for the static and multistage setting, under this umbrella. Indeed, we show that the derived sample sizes estimates in Shapiro(2006) are upper bounds for the sample complexity of the SAA method in the multistage setting. We extend one of this results, relaxing some regularity conditions, to address a more general class of multistage stochastic problems. In our derivation we consider the general (finite) multistage case T ≥ 3 with details. Comparing the upper bounds obtained for the general finite multistage case, T ≥ 3, with the static or two-stage case, T = 2, we observe that, additionally to the exponentially growth behavior with respect to the number of stages, a multiplicative factor of the order (T −1)2(T−1) appears in the derived upper bound for the T−stage case. This shows that the upper bound for T -stage problems grows even faster, with respect to T , than the upper bound for the static case to the power of T − 1.