Abstract In this paper, we study a distributionally robust risk optimization (DRRO) problem where the information on the probability distribution of the underlying random variables is incomplete. But it is possible to use partial information to construct an ambiguity set of probability distributions. In some cases, decision vector x may have a direct impact on the likelihood of the underlying random events that occur after the decision is taken, which motivates us to propose an ambiguity set to be parametric and decision-dependent. To conduct quantitative stability analysis of the optimal value function and the optimal solution mapping of the DRRO problem, we derive error bounds results for the parametrized ambiguity set under the total variation metric and investigate Lipschitz continuity of the objective function of the DRRO problem under some conditions. As an application, we demonstrate that the two-stage stochastic linear semi-definite programs satisfy these conditions and then apply results obtained to it.

中文翻译：

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具有全随机线性半定追索权的两阶段分布鲁棒风险优化问题的定量稳定性

摘要 在本文中，我们研究了一个分布稳健的风险优化 (DRRO) 问题，其中基础随机变量的概率分布信息不完整。但是可以使用部分信息来构建概率分布的模糊集。在某些情况下，决策向量 x 可能对做出决策后发生的潜在随机事件的可能性有直接影响，这促使我们提出一个参数化和决策相关的模糊集。对DRRO问题的最优值函数和最优解映射进行定量稳定性分析，我们推导出总变异度量下参数化模糊集的误差界限结果，并研究了某些条件下 DRRO 问题目标函数的 Lipschitz 连续性。作为一个应用，我们证明了两阶段随机线性半定规划满足这些条件，然后将获得的结果应用于它。