Integrated Conditional Estimation-Optimization

Many real-world optimization problems involve uncertain parameters with probability distributions that can be estimated using contextual feature information. In contrast to the standard approach of first estimating the distribution of uncertain parameters and then optimizing the objective based on the estimation, we propose an integrated conditional estimation-optimization (ICEO) framework that estimates the underlying conditional distribution of the random parameter while considering the structure of the optimization problem. We directly model the relationship between the conditional distribution of the random parameter and the contextual features and then estimate the probabilistic model with an objective that aligns with the downstream optimization problem. We show that our ICEO approach is asymptotically consistent under moderate regularity conditions and further provide finite performance guarantees in the form of generalization bounds. Computationally, performing estimation with the ICEO approach is a nonconvex and often nondifferentiable optimization problem. We propose a general methodology for approximating the potentially nondifferentiable mapping from estimated conditional distribution to the optimal decision by a differentiable function, which greatly improves the performance of gradient-based algorithms applied to the nonconvex problem. We also provide a polynomial optimization solution approach in the semi-algebraic case. Numerical experiments are also conducted to show the empirical success of our approach in different situations, including with limited data samples and model mismatches.

Funding: This work was supported by the Division of Civil, Mechanical, and Manufacturing Innovation [2112533] and the NSF AI Institute for Advances in Optimization Award 2112533 (to P. Grigas).

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2023.0427.