  A key challenge in the emerging field of precision nutrition entails
providing diet recommendations that reflect both the (often unknown) dietary
preferences of different patient groups and known dietary constraints specified
by human experts. Motivated by this challenge, we develop a preference-aware
constrained-inference approach in which the objective function of an
optimization problem is not pre-specified and can differ across various
segments. Among existing methods, clustering models from machine learning are
not naturally suited for recovering the constrained optimization problems,
whereas constrained inference models such as inverse optimization do not
explicitly address non-homogeneity in given datasets. By harnessing the
strengths of both clustering and inverse optimization techniques, we develop a
novel approach that recovers the utility functions of a constrained
optimization process across clusters while providing optimal diet
recommendations as cluster representatives. Using a dataset of patients' daily
food intakes, we show how our approach generalizes stand-alone clustering and
inverse optimization approaches in terms of adherence to dietary guidelines and
partitioning observations, respectively. The approach makes diet
recommendations by incorporating both patient preferences and expert
recommendations for healthier diets, leading to structural improvements in both
patient partitioning and nutritional recommendations for each cluster. An
appealing feature of our method is its ability to consider infeasible but
informative observations for a given set of dietary constraints. The resulting
recommendations correspond to a broader range of dietary options, even when
they limit unhealthy choices.
