Mixed-Integer Optimization with Constraint Learning

We establish a broad methodological foundation for mixed-integer optimization with learned constraints. We propose an end-to-end pipeline for data-driven decision making in which constraints and objectives are directly learned from data using machine learning, and the trained models are embedded in an optimization formulation. We exploit the mixed-integer optimization-representability of many machine learning methods, including linear models, decision trees, ensembles, and multi-layer perceptrons. The consideration of multiple methods allows us to capture various underlying relationships between decisions, contextual variables, and outcomes. We also characterize a decision trust region using the convex hull of the observations, to ensure credible recommendations and avoid extrapolation. We efficiently incorporate this representation using column generation and clustering. In combination with domain-driven constraints and objective terms, the embedded models and trust region define a mixed-integer optimization problem for prescription generation. We implement this framework as a Python package (OptiCL) for practitioners. We demonstrate the method in both chemotherapy optimization and World Food Programme planning. The case studies illustrate the benefit of the framework in generating high-quality prescriptions, the value added by the trust region, the incorporation of multiple machine learning methods, and the inclusion of multiple learned constraints.