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.

我们研究决策相关的分布鲁棒优化模型，其中概率分布的歧义集可以取决于决策变量。这些模型出现在具有内生不确定性的情况下。作为特殊情况，开发的框架包括两阶段决策相关的分布鲁棒随机规划。考虑了五种类型的歧义集的决策相关性概括。这些集合基于矩，协方差矩阵，Wasserstein度量，Phi-散度和Kolmogorov-Smirnov检验的界限。对于有限支持情况，我们使用线性，圆锥或拉格朗日对偶性对这些模型进行了有限数量的约束。还为基于矩，协方差，Wasserstein和Kolmogorov-Smirnov的模型的连续支持情况重新制定了公式。这些重新制定允许在切削面算法的框架内使用全局优化技术解决此类问题。借助一个数值示例，该示例对新闻供应商问题的订单数量和售价的同时确定进行建模，从而证明了决策依赖在歧义集中的重要性。