Inverse Optimization with Noisy Data

TL;DR

This paper develops a statistically consistent inverse optimization method for noisy data in convex problems, introducing a duality-based reformulation, regularization, and two algorithms, validated on synthetic and real datasets.

Contribution

It presents a novel duality-based reformulation and regularization scheme for inverse optimization with noisy data, ensuring statistical consistency and providing two effective solution algorithms.

Findings

The proposed method is statistically consistent under noisy measurements.

Algorithms perform well on synthetic and real data, matching or surpassing existing heuristics.

Regularization allows approximation of the original problem to arbitrary accuracy.

Abstract

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 epi-convergence 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. Lastly, using both synthetic and real data, we demonstrate that our approach performs competitively when compared with existing heuristics.