Inverse Optimization with Noisy Data

Inverse optimization refers to the inference of unknown parameters of an optimization problem based on knowledge of its optimal solutions. This paper considers inverse optimization in the setting where measurements of the optimal solutions of a convex optimization problem are corrupted by noise. We first provide a formulation for inverse optimization and prove it to be NP-hard. In contrast to existing methods, we show that the parameter estimates produced by our formulation are statistically consistent. Our approach involves combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in the formulation. Using epiconvergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, which we use to prove our consistency results. Next, we propose two solution algorithms based on our duality-based formulation. The first is an enumeration algorithm that is applicable to settings where the dimensionality of the parameter space is modest, and the second is a semiparametric approach that combines nonparametric statistics with a modified version of our formulation. These numerical algorithms are shown to maintain the statistical consistency of the underlying formulation. Finally, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics. Funding: The authors gratefully acknowledge the support of the National Science Foundation [Award CMMI-1450963] and a Natural Sciences and Engineering Research Council of Canada Postgraduate Scholarship. This work was partially supported by the National Science Foundation [Grant CMMI-1265671] and the National Science Foundation of China [Grants 71210002 and 71332005]. Keywords: inverse optimization * estimation * statistical learning * semiparametric algorithm
1. Introduction An appreciable share of real-world data represents decisions, which can often be characterized as the solutions of correspondingly defined optimization problems. Estimating the parameters of these latent optimization [...]