# Data-Driven Estimation in Equilibrium Using Inverse Optimization

arXiv: https://arxiv.org/abs/1308.3397 · HTML: https://mlsanity.com/paper/1308.3397/

## Key facts

- Cited by 32 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Aug 2013.

## Built on by (most cited first, 30 of 32)

- [Distributionally Robust Learning](https://mlsanity.com/paper/2108.08993/) (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)
- [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)
- [On Solution Functions of Optimization: Universal Approximation and Covering Number Bounds](https://mlsanity.com/paper/2212.01314/) (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)
- [Estimating and Incentivizing Imperfect-Knowledge Agents with Hidden Rewards](https://mlsanity.com/paper/2308.06717/) (2023)
- [Is Nash Equilibrium Approximator Learnable?](https://mlsanity.com/paper/2108.07472/) (2021)
- [Online Learning for Traffic Routing under Unknown Preferences](https://mlsanity.com/paper/2203.17150/) (2022)
- [Exact Solution to Data-Driven Inverse Optimization of MILPs in Finite Time via Gradient-Based Methods](https://mlsanity.com/paper/2405.14273/) (2024)
- [Conv-FinRe: A Conversational and Longitudinal Benchmark for Utility-Grounded Financial Recommendation](https://mlsanity.com/paper/2602.16990/) (2026)
- [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)
- [Scalable Kernel Inverse Optimization](https://mlsanity.com/paper/2410.23952/) (2024)
- [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)
- [Online Learning for Equilibrium Pricing in Markets under Incomplete Information](https://mlsanity.com/paper/2303.11522/) (2023)
- [Active Inverse Learning in Stackelberg Trajectory Games](https://mlsanity.com/paper/2308.08017/) (2023)
- [Operator Splitting for Learning to Predict Equilibria in Convex Games](https://mlsanity.com/paper/2106.00906/) (2021)
- [An Adaptive Optimization Approach to Personalized Financial Incentives in Mobile Behavioral Weight Loss Interventions](https://mlsanity.com/paper/2307.00444/) (2023)
- [A Fenchel-Young Loss Approach to Data-Driven Inverse Optimization](https://mlsanity.com/paper/2502.16120/) (2025)
- [Inverse Control Constrained Optimization of Vessel Speed Decisions Under Environmental Risk: Evidence from Arctic Shipping](https://mlsanity.com/paper/2605.27270/) (2026)

## Abstract (arXiv)

Equilibrium modeling is common in a variety of fields such as game theory and transportation science. The inputs for these models, however, are often difficult to estimate, while their outputs, i.e., the equilibria they are meant to describe, are often directly observable. By combining ideas from inverse optimization with the theory of variational inequalities, we develop an efficient, data-driven technique for estimating the parameters of these models from observed equilibria. We use this technique to estimate the utility functions of players in a game from their observed actions and to estimate the congestion function on a road network from traffic count data. A distinguishing feature of our approach is that it supports both parametric and nonparametric estimation by leveraging ideas from statistical learning (kernel methods and regularization operators). In computational experiments involving Nash and Wardrop equilibria in a nonparametric setting, we find that a) we effectively estimate the unknown demand or congestion function, respectively, and b) our proposed regularization technique substantially improves the out-of-sample performance of our estimators.