  We consider embedding a predictive machine-learning model within a
prescriptive optimization problem. In this setting, called constraint learning,
we study the concept of a validity domain, i.e., a constraint added to the
feasible set, which keeps the optimization close to the training data, thus
helping to ensure that the computed optimal solution exhibits less prediction
error. In particular, we propose a new validity domain which uses a standard
convex-hull idea but in an extended space. We investigate its properties and
compare it empirically with existing validity domains on a set of test problems
for which the ground truth is known. Results show that our extended convex hull
routinely outperforms existing validity domains, especially in terms of the
function value error, that is, it exhibits closer agreement between the true
function value and the predicted function value at the computed optimal
solution. We also consider our approach within two stylized optimization
models, which show that our method reduces feasibility error, as well as a
real-world pricing case study.
