Sequential Bayesian Risk Set Inference for Robust Discrete Optimization via Simulation

Abstract: Optimization via simulation (OvS) procedures that assume the simulation inputs are generated from the real-world distributions are subject to the risk of selecting a suboptimal solution when the distributions are substituted with input models estimated from finite real-world data -- known as input model risk. Focusing on discrete OvS, this paper proposes a new Bayesian framework for analyzing input model risk of implementing an arbitrary solution, $x$, where uncertainty about the input models is captured by a posterior distribution. We define the $\alpha$-level risk set of solution $x$ as the set of solutions whose expected performance is better than $x$ by a practically meaningful margin $(>\delta)$ given common input models with significant probability ($>\alpha$) under the posterior distribution. The user-specified parameters, $\delta$ and $\alpha$, control robustness of the procedure to the desired level as well as guards against unnecessary conservatism. An empty risk set implies that there is no practically better solution than $x$ with significant probability even though the real-world input distributions are unknown. For efficient estimation of the risk set, the conditional mean performance of a solution given a set of input distributions is modeled as a Gaussian process (GP) that takes the solution-distributions pair as an input. In particular, our GP model allows both parametric and nonparametric input models. We propose the sequential risk set inference procedure that estimates the risk set and selects the next solution-distributions pair to simulate using the posterior GP at each iteration. We show that simulating the pair expected to change the risk set estimate the most in the next iteration is the asymptotic one-step optimal sampling rule that minimizes the number of incorrectly classified solutions, if the procedure runs without stopping.