# Learning in Inverse Optimization: Incenter Cost, Augmented Suboptimality Loss, and Algorithms

arXiv: https://arxiv.org/abs/2305.07730 · HTML: https://mlsanity.com/paper/2305.07730/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 78 references that are in the map.
- First posted on arXiv 12 May 2023.

## Builds on (most cited first, 19 of 19)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [Online Learning: A Modern Introduction Using Convex Optimization](https://mlsanity.com/paper/1912.13213/) (2019)
- [Smart "Predict, then Optimize"](https://mlsanity.com/paper/1710.08005/) (2017)
- [Relatively-Smooth Convex Optimization by First-Order Methods, and Applications](https://mlsanity.com/paper/1610.05708/) (2016)
- [Linear Coupling: An Ultimate Unification of Gradient and Mirror Descent](https://mlsanity.com/paper/1407.1537/) (2014)
- [Generalization Bounds in the Predict-then-Optimize Framework](https://mlsanity.com/paper/1905.11488/) (2019)
- [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)
- [Inverse Optimization: Closed-form Solutions, Geometry and Goodness of fit](https://mlsanity.com/paper/1511.04650/) (2015)
- [Semi-Discrete Optimal Transport: Hardness, Regularization and Numerical Solution](https://mlsanity.com/paper/2103.06263/) (2021)
- [An Inverse Optimization Approach to Measuring Clinical Pathway Concordance](https://mlsanity.com/paper/1906.02636/) (2019)
- [Online Convex Optimization Perspective for Learning from Dynamically Revealed Preferences](https://mlsanity.com/paper/2008.10460/) (2020)
- [Short-term Forecasting of Price-responsive Loads Using Inverse Optimization](https://mlsanity.com/paper/1607.07209/) (2016)
- [Inverse Mixed Integer Optimization: Polyhedral Insights and Trust Region Methods](https://mlsanity.com/paper/2008.00301/) (2020)
- [The Generalized Trust Region Subproblem: solution complexity and convex hull results](https://mlsanity.com/paper/1907.08843/) (2019)

## Built on by (most cited first, 10 of 10)

- [Online Inverse Linear Optimization: Efficient Logarithmic-Regret Algorithm, Robustness to Suboptimality, and Lower Bound](https://mlsanity.com/paper/2501.14349/) (2025)
- [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)
- [Scalable Kernel Inverse Optimization](https://mlsanity.com/paper/2410.23952/) (2024)
- [Offline Reinforcement Learning via Inverse Optimization](https://mlsanity.com/paper/2502.20030/) (2025)
- [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)
- [Tight Regret Bound for Online Inverse Linear Optimization via Multiscale Matrix Weights](https://mlsanity.com/paper/2609.26978/) (2026)
- [Contextual Preference Distribution Learning](https://mlsanity.com/paper/2603.17139/) (2026)

## Abstract (arXiv)

In Inverse Optimization (IO), an expert agent solves an optimization problem parametric in an exogenous signal. From a learning perspective, the goal is to learn the expert's cost function given a dataset of signals and corresponding optimal actions. Motivated by the geometry of the IO set of consistent cost vectors, we introduce the "incenter" concept, a new notion akin to circumcenter recently proposed by Besbes et al. (2023). Discussing the geometric and robustness interpretation of the incenter cost vector, we develop corresponding tractable convex reformulations, which are in contrast with the circumcenter, which we show is equivalent to an intractable optimization program. We further propose a novel loss function called Augmented Suboptimality Loss (ASL), a relaxation of the incenter concept for problems with inconsistent data. Exploiting the structure of the ASL, we propose a novel first-order algorithm, which we name Stochastic Approximate Mirror Descent. This algorithm combines stochastic and approximate subgradient evaluations, together with mirror descent update steps, which is provably efficient for the IO problems with discrete feasible sets with high cardinality. We implement the IO approaches developed in this paper as a Python package called InvOpt. Our numerical experiments are reproducible, and the underlying source code is available as examples in the InvOpt package.