Distributionally robust optimization with decision dependent ambiguity sets

We study decision dependent distributionally robust optimization models, where the ambiguity sets of probability distributions can depend on the decision variables. These models arise in situations with endogenous uncertainty. The developed framework includes two-stage decision dependent distributionally robust stochastic programming as a special case. Decision dependent generalizations of five types of ambiguity sets are considered. These sets are based on bounds on moments, covariance matrix, Wasserstein metric, Phi-divergence and Kolmogorov–Smirnov test. For the finite support case, we use linear, conic or Lagrangian duality to give reformulations of these models with a finite number of constraints. Reformulations are also given for the continuous support case for moment, covariance, Wasserstein and Kolmogorov–Smirnov based models. These reformulations allow solutions of such problems using global optimization techniques within the framework of a cutting surface algorithm. The importance of decision dependence in the ambiguity set is demonstrated with the help of a numerical example modeling simultaneous determination of order quantity and selling price for a newsvendor problem.