# TACIT: Optimization Models that Learn from Their Mistakes

arXiv: https://arxiv.org/abs/2609.38434 · HTML: https://mlsanity.com/paper/2609.38434/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 29 references that are in the map.
- First posted on arXiv 29 Sep 2026.

## Builds on (most cited first, 21 of 21)

- [LLMOPT: Learning to Define and Solve General Optimization Problems from Scratch](https://mlsanity.com/paper/2410.13213/) (2024)
- [Inverse Optimization: Theory and Applications](https://mlsanity.com/paper/2109.03920/) (2021)
- [Autoformulation of Mathematical Optimization Models Using LLMs](https://mlsanity.com/paper/2411.01679/) (2024)
- [OptiMUS-0.3: Using Large Language Models to Model and Solve Optimization Problems at Scale](https://mlsanity.com/paper/2407.19633/) (2024)
- [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)
- [Mixed-Integer Optimization with Constraint Learning](https://mlsanity.com/paper/2111.04469/) (2021)
- [A Survey of Optimization Modeling Meets LLMs: Progress and Future Directions](https://mlsanity.com/paper/2508.10047/) (2025)
- [Optimization with Constraint Learning: A Framework and Survey](https://mlsanity.com/paper/2110.02121/) (2021)
- [Diagnosing Infeasible Optimization Problems Using Large Language Models](https://mlsanity.com/paper/2308.12923/) (2023)
- [Solving General Natural-Language-Description Optimization Problems with Large Language Models](https://mlsanity.com/paper/2407.07924/) (2024)
- [OptimAI: Optimization from Natural Language Using LLM-Powered AI Agents](https://mlsanity.com/paper/2504.16918/) (2025)
- [Contextual Inverse Optimization: Offline and Online Learning](https://mlsanity.com/paper/2106.14015/) (2021)
- [Learning Linear Programs from Optimal Decisions](https://mlsanity.com/paper/2006.08923/) (2020)
- [OptiChat: Bridging Optimization Models and Practitioners with Large Language Models](https://mlsanity.com/paper/2501.08406/) (2025)
- [Inverse Optimization via Learning Feasible Regions](https://mlsanity.com/paper/2505.15025/) (2025)
- [OptiRepair: Closed-Loop Diagnosis and Repair of Supply Chain Optimization Models with LLM Agents](https://mlsanity.com/paper/2602.19439/) (2026)
- [On the Complexity of Inverse Mixed Integer Linear Optimization](https://mlsanity.com/paper/2104.09002/) (2021)
- [Learning Mixed-Integer Linear Programs from Contextual Examples](https://mlsanity.com/paper/2107.07136/) (2021)
- [Inverse Mixed-Integer Programming: Learning Constraints then Objective Functions](https://mlsanity.com/paper/2510.04455/) (2025)
- [Learning Implicit Feasibility Constraints for Real-World Routing and Scheduling: Application to Log Transportation](https://mlsanity.com/paper/2606.05353/) (2026)

## Abstract (arXiv)

Real-world optimization problems are difficult to model accurately because many objectives and constraints reside in domain experts' tacit knowledge, making them hard to formalize. As a result, optimization models often contain miscalibrated objectives, missing constraints, or omitted decision variables, leading to solutions that fail to reflect operational realities. We address this challenge by automatically repairing misspecified formulations using historical data consisting of past solutions and subsequent user overrides. Traditional approaches such as inverse optimization and constraint learning tend to overfit sparse data and produce complex formulations. Our central idea is to combine the reasoning capabilities and prior knowledge of LLMs with the formal grounding provided by optimization. We realize this idea through two complementary paradigms. Top-down, an LLM proposes structural repairs, including new constraints and variables, whose numerical parameters are calibrated and validated through optimization. Bottom-up, optimization infers cuts from observed decisions, which the LLM contextualizes into interpretable, generalizable modeling constraints. We evaluate our approach on 38 misspecification scenarios spanning nine classes of optimization problems, several drawn from real-world applications, and show that TACIT can repair 78.9% of them (vs. 60.5% for the best baseline).