Inverse optimization consists of determining unknown parameters of an optimization problem based on knowledge of its optimal points. This paper considers inverse optimization in the setting where measurements of the optimal points of a convex optimization problem are corrupted by noise. We first provide a formulation for the inverse optimization problem with noisy data and show it is NP-hard, and then we prove that existing convex optimization-based heuristics for inverse optimization with noisy data are statistically inconsistent (meaning the answers provided by these methods generally converge to the "wrong" answer as more data is collected). Next, we show that our formulation is statistically consistent by combining a new duality-based reformulation for bilevel programs with a regularization scheme that smooths discontinuities in our formulation. Using epi-convergence theory, we show the regularization parameter can be adjusted to approximate the original inverse optimization problem to arbitrary accuracy, and this is used to prove our statistical consistency results. This duality-based reformulation is next used to propose two numerical algorithms for solving the inverse optimization problem with noisy data. 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 of the inverse optimization problem. These numerical algorithms are shown to maintain the statistical consistency of our formulation. Lastly, we demonstrate using synthetic and real data sets the competitiveness of our numerical algorithms as compared to existing heuristics.
Further, inspired by REF , we are able to show the statistical consistency of the estimation, which most of the papers that consider noisy observations do not cover.
Inverse Optimization with Noisy Data
arXiv: Optimization and Control
Mathematics Computer Science