# Quantile Inverse Optimization: Improving Stability in Inverse Linear Programming

## TL;DR

This paper introduces a new inverse linear programming method focused on stability, addressing sensitivity issues in data-driven inverse LP by ensuring solutions are robust to data imperfections and applicable in real-world scenarios.

## Contribution

The paper proposes a novel inverse LP approach that guarantees stability under data noise, formulates it as a mixed-integer program, and develops heuristics for efficient solutions.

## Key findings

- Enhanced stability of inverse LP solutions under data noise
- Effective application in diet and transshipment problems
- Development of heuristics for large-scale problems

## Abstract

Inverse linear programming (LP) has received increasing attention due to its potential to generate efficient optimization formulations that can closely replicate the behavior of a complex system. However, inversely inferred parameters and corresponding forward solutions from the existing inverse LP method can be highly sensitive to noise, errors, and uncertainty in the input data, limiting its applicability in data-driven settings. We introduce the notion of inverse and forward stability in inverse LP and propose a novel inverse LP method that determines a set of objective functions that are stable under data imperfection and generate solutions close to the relevant subset of the data. We formulate the inverse model as a mixed-integer program and elucidate its connection to bi-clique problems, which we exploit to develop efficient heuristics. We also show how this method can be used for online learning. We numerically evaluate the stability of the proposed method and demonstrate its practical use in the diet recommendation and transshipment applications.