# Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions

arXiv: https://arxiv.org/abs/2510.04455 · HTML: https://mlsanity.com/paper/2510.04455/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 43 references that are in the map.
- First posted on arXiv 6 Oct 2025.

## Builds on (most cited first, 16 of 16)

- [Inverse Optimization: Theory and Applications](https://mlsanity.com/paper/2109.03920/) (2021)
- [Inverse Optimization with Noisy Data](https://mlsanity.com/paper/1507.03266/) (2015)
- [Data-driven Inverse Optimization with Imperfect Information](https://mlsanity.com/paper/1512.05489/) (2015)
- [Data-Driven Estimation in Equilibrium Using Inverse Optimization](https://mlsanity.com/paper/1308.3397/) (2013)
- [Contextual Inverse Optimization: Offline and Online Learning](https://mlsanity.com/paper/2106.14015/) (2021)
- [An Online-Learning Approach to Inverse Optimization](https://mlsanity.com/paper/1810.12997/) (2018)
- [Contextual Recommendations and Low-Regret Cutting-Plane Algorithms](https://mlsanity.com/paper/2106.04819/) (2021)
- [Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit](https://mlsanity.com/paper/1511.04650/) (2015)
- [Learning in Inverse Optimization: Incenter Cost, Augmented Suboptimality Loss, and Algorithms](https://mlsanity.com/paper/2305.07730/) (2023)
- [Inverse optimization for the recovery of constraint parameters](https://mlsanity.com/paper/1811.00726/) (2018)
- [An Inverse Optimization Approach to Measuring Clinical Pathway Concordance](https://mlsanity.com/paper/1906.02636/) (2019)
- [Inverse Optimization via Learning Feasible Regions](https://mlsanity.com/paper/2505.15025/) (2025)
- [Automating Personnel Rostering by Learning Constraints Using Tensors](https://mlsanity.com/paper/1805.11375/) (2018)
- [Exact Solution to Data-Driven Inverse Optimization of MILPs in Finite Time via Gradient-Based Methods](https://mlsanity.com/paper/2405.14273/) (2024)
- [A proof of convergence of inverse reinforcement learning for multi-objective optimization](https://mlsanity.com/paper/2305.06137/) (2023)
- [A proof of imitation of Wasserstein inverse reinforcement learning for multi-objective optimization](https://mlsanity.com/paper/2305.10089/) (2023)

## Built on by (most cited first, 2 of 2)

- [TACIT: Optimization Models that Learn from Their Mistakes](https://mlsanity.com/paper/2609.38434/) (2026)
- [Outperformance Inverse Optimization: Learning Objective Functions that Outperform Agent Decisions](https://mlsanity.com/paper/2610.09890/) (2026)

## Abstract (arXiv)

Data-driven inverse optimization for mixed-integer linear programs (MILPs), which seeks to learn an objective function and constraints consistent with observed decisions, is important for building accurate mathematical models in a variety of domains, including power systems and scheduling. However, to the best of our knowledge, existing data-driven inverse optimization methods primarily focus on learning objective functions under known constraints, and learning both objective functions and constraints from data for MILPs remains largely unexplored. In this paper, we propose a two-stage approach for a class of inverse optimization problems in which the objective is a linear combination of given feature functions and the constraints are parameterized by unknown functions and thresholds. Our method first learns the constraints and then, conditioned on the learned constraints, estimates the objective-function weights. On the theoretical side, we provide finite-sample guarantees for solving the proposed inverse optimization problem. To this end, we develop statistical learning tools for pseudo-metric spaces under sub-Gaussian assumptions and use them to derive a learning-theoretic framework for inverse optimization with both unknown objectives and constraints. On the experimental side, we demonstrate that our method successfully solves inverse optimization problems on scheduling instances formulated as ILPs with up to 100 decision variables.