On Distributionally Robust Multiperiod Stochastic Optimization

Abstract

This paper considers model uncertainty for multistage stochastic programs. The data and information structure of the baseline model is a tree, on which the decision problem is defined. We consider ambiguity neighborhoods around this tree as alternative models which are close to the baseline model. Closeness is defined in terms of a distance for probability trees, called the nested distance. This distance is appropriate for scenario models of multistage stochastic optimization problems as was demonstrated in (Pflug and Pichler, 2012). The ambiguity model is formulated as a minimax problem, where the optimal decision is to be found, which minimizes the maximal objective function, within the ambiguity set. We give a setup for studying saddle point properties of the minimax problem. Moreover, we present solution algorithms for finding the minimax decisions at least asymptotically. As an example, we consider a multiperiod stochastic production/inventory control problem with weekly ordering. The stochastic scenario process is given by the random demands for two products. We find the worst trees within the ambiguity set and determine a solution which is robust w.r.t. model uncertainty. It turns out that the probability weights of the worst case trees are concentrated on few very bad scenarios.