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

Farzin Ahmadi, Fardin Ganjkhanloo, Kimia Ghobadi

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.

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

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.
Paper Structure (40 sections, 30 theorems, 22 equations, 18 figures, 11 tables, 1 algorithm)

This paper contains 40 sections, 30 theorems, 22 equations, 18 figures, 11 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Relevant Literature
  4. Inverse Optimization for Parametric Convex Models
  5. Preliminaries
  6. Non-Identifiability of $\mathcal{ICO}$ and $\mathcal{R}$-$\mathcal{ICO}$
  7. Sufficient Conditions for Parameter Identifiability
  8. Inverse Learning: A Scalable Framework for Data-Rich Inverse Optimization
  9. The Inverse Learning Formulation
  10. Characterization of $\mathcal{IL}$ Properties
  11. Identifiability of Inverse Learning
  12. Goal-Integrated Inverse Learning: Navigating the Observation-Constraint Tradeoff
  13. Goal-Integrated Inverse Learning (GIL)
  14. Theoretical Properties of $\mathcal{GIL}$
  15. Modified Goal-Integrated Inverse Learning ($\mathcal{MGIL}$)
  16. ...and 25 more sections

Key Result

proposition 1

Let $\{z^k\}_{k=1}^K\subseteq\Omega$ be imputed optima. Define the common normal cone intersection Then $\mathcal{C}$ is a polyhedral cone. Moreover, let $I_\cap := \bigcap_{k=1}^K I(z^k)$ denote the common active set. Then:

Figures (18)

  • Figure 1: Geometric Illustration of non-identifiability in inverse linear optimization. (a) At a vertex $z^1$ where two non-collinear constraints are active, the normal cone $N_\Omega(z^1)$ is two-dimensional, admitting multiple feasible parameters $\theta^{(1)}, \theta^{(2)} \in N_\Omega(z^1) \cap \Theta$ (Proposition \ref{['prop:multiple-rays-conditions']}(ii)). (b) With two observations $z^1, z^2$, the feasible parameter set is the intersection $\mathcal{C} = N_\Omega(z^1) \cap N_\Omega(z^2)$. Non-identifiability persists unless observations reduce $\mathcal{C}$ to a single ray (Theorem \ref{['thm:nonidentifiability-lp']}).
  • Figure 2: Data generation under the two experimental scenarios. (Left) IL Assumption Scenario: noise is added to a single true optimal solution $x^*$, consistent with Assumption \ref{['assump:IL-data']}. (Right) IO Assumption Scenario: noise is added to potentially different optimal solutions $x^*_k$ for a single true parameter $\theta^*$.
  • Figure 3: Solution distance comparison ($\ell_2$-norm to true solution) under the $\mathcal{IL}$ Assumption Scenario ($n=10$). Boxplots show distributions across 100 instances for varying noise levels (columns) and true solution binding levels (rows). Models include $\mathcal{IL}$, $\mathcal{GIL}(r)$, $\mathcal{MGIL}(r)$ with $r \in \{5, 10\}$, and the classical benchmark $\mathcal{ILO}$. The $\mathcal{IL}$ framework models consistently yield smaller distances than $\mathcal{ILO}$. Distance increases with $r$, illustrating the observation-constraint tradeoff.
  • Figure 4: Solution distance comparison under the $\mathcal{IO}$ Assumption Scenario ($n=10$). Structure and interpretation mirror Figure \ref{['fig:ex2_distances']}. $\mathcal{IL}$ framework models maintain their proximity advantage.
  • Figure 5: Parameter recovery rate as a function of the binding parameter $r$ across knowledge levels (rows) and noise levels (columns). Flat lines represent $\mathcal{IL}$ and $\mathcal{ILO}$ baselines (independent of $r$). The goal-integrated models $\mathcal{GIL}$ and $\mathcal{MGIL}$ (with and without preferred constraints, denoted "NK") achieve monotonically increasing recovery rates with $r$, substantially outperforming the baselines. Preferred constraint knowledge provides consistent additional benefit.
  • ...and 13 more figures

Theorems & Definitions (67)

  • remark 1: Statistical Consistency
  • remark 2: Computational Complexity
  • proposition 1: Normal Cone Intersection Structure
  • proposition 2: Conditions for Multiple Rays in $\mathcal{C}$
  • theorem 1: Non-Identifiability of $\mathcal{ICO}$ in Linear Programs
  • corollary 1: Persistence of Non-Identifiability Under Regularization
  • proposition 3: Feasibility Set Structure for Convex Models
  • proposition 4: Conditions for Non-Identifiability in Convex Models
  • theorem 2: Global Non-Identifiability in Convex Inverse Optimization
  • remark 3: Sources of Ambiguity in Convex Models
  • ...and 57 more