This paper considers data-driven chance-constrained stochastic optimization problems in a Bayesian framework. Bayesian posteriors afford a principled mechanism to incorporate data and prior knowledge into stochastic optimization problems. However, the computation of Bayesian posteriors is typically an intractable problem, and has spawned a large literature on approximate Bayesian computation. Here, in the context of chance-constrained optimization, we focus on the question of statistical consistency (in an appropriate sense) of the optimal value, computed using an approximate posterior distribution. To this end, we rigorously prove a frequentist consistency result demonstrating the weak convergence of the optimal value to the optimal value of a fixed, parameterized constrained optimization problem. We augment this by also establishing a probabilistic rate of convergence of the optimal value. We also prove the convex feasibility of the approximate Bayesian stochastic optimization problem. Finally, we demonstrate the utility of our approach on an optimal staffing problem for an M/M/c queueing model.

翻译：本文探讨了巴伊西亚框架中受数据驱动的受机会限制的随机优化问题。 巴伊西亚后辈提供了一种原则性机制,将数据和先前的知识纳入到随机优化问题中。然而,计算巴伊西亚后辈通常是一个棘手的问题,并产生了大量关于巴伊西亚计算方法的文献。在这里,在受机会限制的优化背景下,我们侧重于最佳价值的统计一致性问题(适当意义上的),使用近似后遗分布进行计算。为此,我们严格地证明,经常出现一致的结果表明,最佳价值与固定的、有参数限制的优化问题的最佳价值的趋同不力。我们还通过确定最佳价值的概率趋同率来增加这一结果。我们还证明了近似巴伊西亚的随机优化问题具有共性的可行性。最后,我们展示了我们对M/M/c类排队列模式的最佳人员配置问题的实用性。