WB10 - Stochastic and Robust Optimization 4

We consider a black-box that, given any input, returns a random variable with unknown distribution. We introduce a methodology that predicts the distributions of all random variables from a set of inputs at which some observations of the associated random variable have been computed. Our methodology also quantifies the uncertainty associated to these predictions. Then we discuss how to select relevant new points at which observations may be computed to refine the global prediction. Finally, we connect this methodology with a stochastic optimization context where the goal is to minimize the expectation of the distribution returned by the stochastic black-box.

Bayesian optimization (BO) commonly employs Gaussian processes (GPs) as surrogate models, but standard GP surrogates with localized kernels often struggle with nonstationary and multimodal objectives, especially over search spaces containing mixed-type inputs such as continuous, integer, and categorical variables. We propose Bayesian Kernelized Tensor Factorization (BKTF), a fully Bayesian surrogate model for BO on Cartesian product spaces with heterogeneous inputs. BKTF represents the objective function using a Bayesian probabilistic kernelized low-rank decomposition, placing GP priors on the latent basis functions for continuous variables to encode consistency and smoothness, and multivariate Gaussian priors on the latent factors for discrete inputs. This formulation allows information from each evaluation to be shared not only with neighbors but also across dimensions, fostering a more global and data-adaptive search strategy that can capture nonstationary and nonseparable structure. We develop an efficient Markov chain Monte Carlo (MCMC) algorithm for posterior inference and construct fully Bayesian Monte Carlo acquisition functions (AFs) from the resulting predictive distribution. Numerical experiments on a nonstationary synthetic function, eight standard BO benchmark functions, and two hyperparameter tuning tasks demonstrate that BKTF offers a competitive and effective surrogate for optimizing complex mixed-variable objectives, particularly when the initial design is small and the evaluation budget is limited.

Retailers frequently make inventory decisions using sales data that are censored by stockouts and distorted by customer substitution, which can bias demand estimation and lead to suboptimal replenishment policies. We propose an integrated estimation-prediction-optimization framework for dynamic inventory management with censored demand and substitution. Our approach decomposes demand into a covariate-driven customer arrival process and a stockout-based substitution choice model, and estimates the resulting latent demand structure through a Bayesian hierarchical framework. To address the high-dimensional latent trajectory problem, we develop a blocked Markov chain Monte Carlo (MCMC) procedure with particle Gibbs and ancestor sampling. The estimated model is then used to generate posterior predictive arrivals and simulate future sales under alternative inventory policies, enabling stochastic optimization of replenishment decisions. The framework provides a coherent way to correct demand bias from stockouts, capture substitution behavior, and support more informed inventory planning.