# Data-driven Inverse Optimization with Imperfect Information

arXiv: https://arxiv.org/abs/1512.05489 · HTML: https://mlsanity.com/paper/1512.05489/

## Key facts

- Cited by 34 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Dec 2015.

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

- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (2021)
- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Contextual Inverse Optimization: Offline and Online Learning](https://mlsanity.com/paper/2106.14015/) (2021)
- [Maximum Optimality Margin: A Unified Approach for Contextual Linear Programming and Inverse Linear Programming](https://mlsanity.com/paper/2301.11260/) (2023)
- [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)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Revisiting Online Learning Approach to Inverse Linear Optimization: A Fenchel-Young Loss Perspective and Gap-Dependent Regret Analysis](https://mlsanity.com/paper/2501.13648/) (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)
- [Coresets for Wasserstein Distributionally Robust Optimization Problems](https://mlsanity.com/paper/2210.04260/) (2022)
- [Simple Projection-Free Algorithm for Contextual Recommendation with Logarithmic Regret and Robustness](https://mlsanity.com/paper/2603.20826/) (2026)
- [Tight Generalization Bounds for Noiseless Inverse Optimization](https://mlsanity.com/paper/2605.08866/) (2026)
- [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)
- [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)
- [From Inverse Optimization to Feasibility to ERM](https://mlsanity.com/paper/2402.17890/) (2024)
- [BO4IO: A Bayesian optimization approach to inverse optimization with uncertainty quantification](https://mlsanity.com/paper/2405.17875/) (2024)
- [Offline Reinforcement Learning via Inverse Optimization](https://mlsanity.com/paper/2502.20030/) (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)
- [On Tractability, Complexity, and Mixed-Integer Convex Programming Representability of Distributionally Favorable Optimization](https://mlsanity.com/paper/2401.17899/) (2024)
- [Estimating the Unobservable Components of Electricity Demand Response with Inverse Optimization](https://mlsanity.com/paper/2410.02774/) (2024)
- [A Fenchel-Young Loss Approach to Data-Driven Inverse Optimization](https://mlsanity.com/paper/2502.16120/) (2025)
- [Contextual Preference Distribution Learning](https://mlsanity.com/paper/2603.17139/) (2026)

## Abstract (arXiv)

In data-driven inverse optimization an observer aims to learn the preferences of an agent who solves a parametric optimization problem depending on an exogenous signal. Thus, the observer seeks the agent's objective function that best explains a historical sequence of signals and corresponding optimal actions. We focus here on situations where the observer has imperfect information, that is, where the agent's true objective function is not contained in the search space of candidate objectives, where the agent suffers from bounded rationality or implementation errors, or where the observed signal-response pairs are corrupted by measurement noise. We formalize this inverse optimization problem as a distributionally robust program minimizing the worst-case risk that the em predicted decision (em i.e., the decision implied by a particular candidate objective) differs from the agent's em actual response to a random signal. We show that our framework offers rigorous out-of-sample guarantees for different loss functions used to measure prediction errors and that the emerging inverse optimization problems can be exactly reformulated as (or safely approximated by) tractable convex programs when a new suboptimality loss function is used. We show through extensive numerical tests that the proposed distributionally robust approach to inverse optimization attains often better out-of-sample performance than the state-of-the-art approaches.