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.

We are working with a company on a hard industrial optimisation problem: a version of the well-known Cutting Stock Problem in which a paper mill must cut rolls of paper following certain cutting patterns to meet customer demands. In our problem each roll to be cut may have a different size, the cutting patterns are semi-automated so that we have only indirect control over them via a list of continuous parameters called a request, and there are multiple mills each able to use only one request. We solve the problem using a combination of machine learning and optimisation techniques. First we approximate the distribution of cutting patterns via Monte Carlo simulation. Secondly we cover the distribution by applying a k-medoids algorithm. Thirdly we use the results to build an ILP model which is then solved.

We provide a framework for optimization with constraint learning.• We review the literature using constraint learning in light of our proposed framework.• Current trends are discussed and opportunities for future work are highlighted.

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