Deterministic branch-and-bound algorithms for continuous global optimization often visit a large number of boxes in the neighborhood of a global minimizer, resulting in the so-called cluster problem (J Glob Optim 5(3):253-265, 1994). This article extends previous analyses of the cluster problem in unconstrained global optimization (
We consider data-driven approaches that integrate a machine learning prediction model within distributionally robust optimization (DRO) given limited joint observations of uncertain parameters and covariates. Our framework is flexible in the sense that it can accommodate a variety of learning setups and DRO ambiguity sets. We investigate the asymptotic and finite sample properties of solutions obtained using Wasserstein, sample robust optimization, and phi-divergence-based ambiguity sets within our DRO formulations, and explore cross-validation approaches for sizing these ambiguity sets. Through numerical experiments, we validate our theoretical results, study the effectiveness of our approaches for sizing ambiguity sets, and illustrate the benefits of our DRO formulations in the limited data regime even when the prediction model is misspecified.
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