From Non-Identifiability to Goal-Integrated Decision-Making in Parametric Inverse Optimization

TL;DR

This paper develops a theoretical framework for inverse optimization, revealing inherent non-identifiability issues, and introduces goal-integrated inverse learning to improve parameter recovery and decision-making in convex models.

Contribution

It introduces the Inverse Learning framework that shifts focus from parameters to solutions, reducing complexity and improving recovery in inverse optimization.

Findings

Non-identifiability is the typical case in inverse optimization.

Inverse Learning reduces complexity and improves parameter recovery.

Numerical experiments show better accuracy and speed, applied to dietary recommendations.

Abstract

Inverse optimization seeks to recover unknown objective parameters from observed decisions, yet fundamental questions about when recovery is possible have received limited formal treatment. This paper develops a comprehensive theoretical framework for inverse optimization in parametric convex models. We first establish that non-identifiability is the generic case: even with normalization and multiple observations, the parameter set compatible with data is generically multi-dimensional, and regularization does not resolve this. We derive necessary and sufficient conditions for identifiability. Motivated by these negative results, we introduce the Inverse Learning (IL) framework, which shifts the inferential target from the unknown parameter to the latent optimal solution, achieving a complexity reduction that is independent of the number of observations. IL explicitly characterizes the full set of compatible parameters rather than returning an arbitrary element. To address the tension between observational fidelity and constraint adherence, we formalize the Observation-Constraint Tradeoff and develop Goal-Integrated Inverse Learning models that enable structured navigation of this spectrum with guaranteed monotonicity. Numerical experiments demonstrate superior solution accuracy, higher parameter recovery rates, and significant computational speedups. We apply the framework to personalized dietary recommendations using NHANES data, proof-of-concept demonstrating improved glycemic control in a prospective feasibility study.