# Inverse Optimization with Noisy Data

arXiv: https://arxiv.org/abs/1507.03266 · HTML: https://mlsanity.com/paper/1507.03266/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jul 2015.

## Built on by (most cited first, 30 of 34)

- [Contextual Inverse Optimization: Offline and Online Learning](https://mlsanity.com/paper/2106.14015/) (2021)
- [Learning in Inverse Optimization: Incenter Cost, Augmented Suboptimality Loss, and Algorithms](https://mlsanity.com/paper/2305.07730/) (2023)
- [Online Inverse Linear Optimization: Efficient Logarithmic-Regret Algorithm, Robustness to Suboptimality, and Lower Bound](https://mlsanity.com/paper/2501.14349/) (2025)
- [Finite and Corruption-Robust Regret Bounds in Online Inverse Linear Optimization under M-Convex Action Sets](https://mlsanity.com/paper/2602.01682/) (2026)
- [Repeated Principal-Agent Games with Unobserved Agent Rewards and Perfect-Knowledge Agents](https://mlsanity.com/paper/2304.07407/) (2023)
- [Tight Generalization Bounds for Noiseless Inverse Optimization](https://mlsanity.com/paper/2605.08866/) (2026)
- [Regret Analysis of Learning-Based MPC with Partially-Unknown Cost Function](https://mlsanity.com/paper/2108.02307/) (2021)
- [Inverse Optimization for Routing Problems](https://mlsanity.com/paper/2307.07357/) (2023)
- [Estimating and Incentivizing Imperfect-Knowledge Agents with Hidden Rewards](https://mlsanity.com/paper/2308.06717/) (2023)
- [Data-driven decision-focused surrogate modeling](https://mlsanity.com/paper/2308.12161/) (2023)
- [Exact Solution to Data-Driven Inverse Optimization of MILPs in Finite Time via Gradient-Based Methods](https://mlsanity.com/paper/2405.14273/) (2024)
- [Using Inverse Optimization to Learn Cost Functions in Generalized Nash Games](https://mlsanity.com/paper/2102.12415/) (2021)
- [Efficient Learning of Decision-Making Models: A Penalty Block Coordinate Descent Algorithm for Data-Driven Inverse Optimization](https://mlsanity.com/paper/2210.15393/) (2022)
- [BO4IO: A Bayesian optimization approach to inverse optimization with uncertainty quantification](https://mlsanity.com/paper/2405.17875/) (2024)
- [Scalable Kernel Inverse Optimization](https://mlsanity.com/paper/2410.23952/) (2024)
- [Conformalized Decision Risk Assessment](https://mlsanity.com/paper/2505.13243/) (2025)
- [Efficient Online Inverse Optimization with O(d) Regret](https://mlsanity.com/paper/2609.13440/) (2026)
- [Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions](https://mlsanity.com/paper/2510.04455/) (2025)
- [Explicit Iteration Complexity of Exact Data-Driven Inverse Optimization for Integer Linear Programs](https://mlsanity.com/paper/2607.22263/) (2026)
- [Tight Regret Bound for Online Inverse Linear Optimization via Multiscale Matrix Weights](https://mlsanity.com/paper/2609.26978/) (2026)
- [Inverse Bayesian Optimization: Learning Human Acquisition Functions in an Exploration vs Exploitation Search Task](https://mlsanity.com/paper/2104.09237/) (2021)
- [Estimating the Unobservable Components of Electricity Demand Response with Inverse Optimization](https://mlsanity.com/paper/2410.02774/) (2024)
- [Learning Pareto manifolds in high dimensions: How can regularization help?](https://mlsanity.com/paper/2503.08849/) (2025)
- [Improving Sample Efficiency of Deep Learning Models in Electricity Market](https://mlsanity.com/paper/2210.05599/) (2022)
- [Modeling Non-deterministic Human Behaviors in Discrete Food Choices](https://mlsanity.com/paper/2301.09454/) (2023)
- [Follower Agnostic Methods for Stackelberg Games](https://mlsanity.com/paper/2302.01421/) (2023)
- [An Adaptive Optimization Approach to Personalized Financial Incentives in Mobile Behavioral Weight Loss Interventions](https://mlsanity.com/paper/2307.00444/) (2023)
- [Inverse Optimization Latent Variable Models for Learning Costs Applied to Route Problems](https://mlsanity.com/paper/2509.15999/) (2025)
- [Behavior Learning (BL): Learning Hierarchical Optimization Structures from Data](https://mlsanity.com/paper/2602.20152/) (2026)
- [Inverse Control Constrained Optimization of Vessel Speed Decisions Under Environmental Risk: Evidence from Arctic Shipping](https://mlsanity.com/paper/2605.27270/) (2026)

## Abstract (arXiv)

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.