# Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit

arXiv: https://arxiv.org/abs/1511.04650 · HTML: https://mlsanity.com/paper/1511.04650/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Nov 2015.

## Built on by (most cited first, 13 of 13)

- [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)
- [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)
- [Repeated Principal-Agent Games with Unobserved Agent Rewards and Perfect-Knowledge Agents](https://mlsanity.com/paper/2304.07407/) (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)
- [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)
- [Inverse Bayesian Optimization: Learning Human Acquisition Functions in an Exploration vs Exploitation Search Task](https://mlsanity.com/paper/2104.09237/) (2021)
- [A Fenchel-Young Loss Approach to Data-Driven Inverse Optimization](https://mlsanity.com/paper/2502.16120/) (2025)

## Abstract (arXiv)

In classical inverse linear optimization, one assumes a given solution is a candidate to be optimal. Real data is imperfect and noisy, so there is no guarantee this assumption is satisfied. Inspired by regression, this paper presents a unified framework for cost function estimation in linear optimization comprising a general inverse optimization model and a corresponding goodness-of-fit metric. Although our inverse optimization model is nonconvex, we derive a closed-form solution and present the geometric intuition. Our goodness-of-fit metric, ρ, the coefficient of complementarity, has similar properties to R2 from regression and is quasiconvex in the input data, leading to an intuitive geometric interpretation. While ρ is computable in polynomial-time, we derive a lower bound that possesses the same properties, is tight for several important model variations, and is even easier to compute. We demonstrate the application of our framework for model estimation and evaluation in production planning and cancer therapy.