Inverse optimization has been increasingly used to estimate unknown parameters in an optimization model based on decision data. We show that such a point estimation is insufficient in a prescriptive setting where the estimated parameters are used to prescribe new decisions. The prescribed decisions may be low-quality and misaligned with human intuition and thus are unlikely to be adopted. To tackle this challenge, we propose conformal inverse optimization, which seeks to learn an uncertainty set for the unknown parameters and then solve a robust optimization model to prescribe new decisions. Under mild assumptions, we show that the suggested decisions can achieve bounded out-of-sample optimality gaps, as evaluated using both the ground-truth parameters and the decision maker's perception of the unknown parameters. Our method demonstrates strong empirical performance compared to classic inverse optimization.

We propose a data-driven technique to automatically learn the uncertainty sets in robust optimization. Our method reshapes the uncertainty sets by minimizing the expected performance across a family of problems subject to guaranteeing constraint satisfaction. Our approach is very flexible and can learn a wide variety of uncertainty sets while preserving tractability. We solve the constrained learning problem using a stochastic augmented Lagrangian method that relies on differentiating the solutions of the robust optimization problems with respect to the parameters of the uncertainty set. Due to the nonsmooth and nonconvex nature of the augmented Lagrangian function, we apply the nonsmooth conservative implicit function theorem to establish convergence to a critical point, which is a feasible solution of the constrained problem under mild assumptions. Using empirical process theory, we show finite-sample probabilistic guarantees of constraint satisfaction for the resulting solutions. Numerical experiments show that our method outperforms traditional approaches in robust and distributionally robust optimization in terms of out-of-sample performance and constraint satisfaction guarantees.

Robust optimization provides a mathematical framework for modeling and solving decision-making problems under worst-case uncertainty. This work addresses two-stage robust optimization (2RO) problems (also called adjustable robust optimization), wherein first-stage and second-stage decisions are made before and after uncertainty is realized, respectively. This results in a nested min-max-min optimization problem which is extremely challenging computationally, especially when the decisions are discrete. We propose Neur2RO, an efficient machine learning-driven instantiation of column-and-constraint generation (CCG), a classical iterative algorithm for 2RO. Specifically, we learn to estimate the value function of the second-stage problem via a novel neural network architecture that is easy to optimize over by design. Embedding our neural network into CCG yields high-quality solutions quickly as evidenced by experiments on two 2RO benchmarks, knapsack and capital budgeting. For knapsack, Neur2RO finds solutions that are within roughly 2\% of the best-known values in a few seconds compared to the three hours of the state-of-the-art exact branch-and-price algorithm; for larger and more complex instances, Neur2RO finds even better solutions. For capital budgeting, Neur2RO outperforms three variants of the k-adaptability algorithm, particularly on the largest instances, with a 5 to 10-fold reduction in solution time.