Bayesian Joint Chance Constrained Optimization: Approximations and Statistical Consistency
This addresses the problem of reliable data-driven decision-making under uncertainty for operations research and stochastic optimization, but it is incremental as it builds on existing Bayesian approximation methods.
The paper tackles the challenge of ensuring statistical consistency in Bayesian chance-constrained optimization when using approximate posteriors, proving convergence of the optimal value to a fixed parameterized problem and demonstrating it on an M/M/c queueing staffing problem.
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 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.