A distance for multistage stochastic optimization models (Q2902866)

The observations in a multistage stochastic programming problem are modeled as a stochastic scenario process, and a decision may depend on the observed values. Therefore, a method of optimization in functional spaces is needed to find the optimal decision. The authors have constructed a space for the presentation of the outcomes of a process and the evolving information. The concept is based on the nested distributions which describe the scenario values together with the information structure which aids the decision making. The nested distance between these distributions is introduced by the generalization of the Wasserstein distance which is known as useful in the analysis of two-stage problems. The main result states that the difference of the optimal values of two multistage stochastic programs, which are Lipshitz and differ only in nested distributions of the stochastic parameters, can be bounded by the nested distance of these distributions. A theorem on the dual representation of the multistage distances is proven. The results of the paper are applicable for general stochastic processes as well as for finite scenario trees.