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, which allows us to capture various underlying relationships between decisions, contextual variables, and outcomes. We also introduce two approaches for handling the inherent uncertainty of learning from data. First, we 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 propose a more flexible formulation to deal with low-density regions and high-dimensional datasets. Then, we propose an ensemble learning approach that enforces constraint satisfaction over multiple bootstrapped estimators or multiple algorithms. In combination with domain-driven components, 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 World Food Programme planning and chemotherapy optimization. The case studies illustrate the framework's ability to generate high-quality prescriptions as well as the value added by the trust region, the use of ensembles to control model robustness, the consideration of multiple machine learning methods, and the inclusion of multiple learned constraints

Using Limited Trial Evidence to Credibly Choose Treatment Dosage when Efficacy and Adverse Effects Weakly Increase with Dose

In medical treatment and elsewhere, it has become standard to base treatment intensity (dosage) on evidence in randomized trials. Yet it has been rare to study how outcomes vary with dosage. In trials to obtain drug approval, the norm has been to specify some dose of a new drug and compare it with an established therapy or placebo. Design-based trial analysis views each trial arm as qualitatively different, but it may be highly credible to assume that efficacy and adverse effects (AEs) weakly increase with dosage. Optimization of patient care requires joint attention to both, as well as to treatment cost. This paper develops methodology to credibly use limited trial evidence to choose dosage when efficacy and AEs weakly increase with dose. I suppose that dosage is an integer choice t in (0, 1, . . . , T), T being a specified maximum dose. I study dosage choice when trial evidence on outcomes is available for only K dose levels, where K < T + 1. Then the population distribution of dose response is partially rather than point identified. The identification region is a convex polygon determined by linear equalities and inequalities. I characterize clinical and public-health decision making using the minimax-regret criterion. A simple analytical solution exists when T = 2 and computation is tractable when T is larger