Automated Optimization Modeling via a Localizable Error-Driven Perspective

Weiting Liu; Han Wu; Yufei Kuang; Xiongwei Han; Tao Zhong; Jianfeng Feng; Wenlian Lu

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Automated Optimization Modeling via a Localizable Error-Driven Perspective

Weiting Liu, Han Wu, Yufei Kuang, Xiongwei Han, Tao Zhong, Jianfeng Feng, Wenlian Lu

arXiv:2602.11164v1Jan 17, 2026cs.LGcs.AIcs.CL

TL;DR

Automated optimization modeling post-training is limited by sparse error-specific problems and sparse rewards on difficult tasks. MIND addresses both through localized error-driven data synthesis and DFPO, and consistently outperforms state-of-the-art methods across six benchmarks. The paper also identifies a reward-design assumption that supports its error-based training signal.

Problem
Automated optimization modeling post-training lacks sufficient error-specific training problems and learning signals for difficult problems.
Method
MIND combines an error-driven reverse data synthesis pipeline with DFPO, which dynamically corrects wrong responses while keeping corrected responses near the base model response distribution.
Results
MIND consistently outperforms state-of-the-art automated optimization modeling methods across six benchmarks.
Takeaways & Limitations
Modeling errors are often localized to semantic segments, supporting focused high-density data synthesis and localized post-training for automated optimization modeling.
Takeaways & Limitations
The reward design assumes that the objective value of a mathematical formulation can support error-based guidance toward structural closeness to the correct formulation.

Abstract

from arXiv · show

Automated optimization modeling via Large Language Models (LLMs) has emerged as a promising approach to assist complex human decision-making. While post-training has become a pivotal technique to enhance LLMs' capabilities in this domain, its effectiveness is severely constrained by the scarcity and underutilization of high-quality training data. However, through a detailed profiling of error patterns across various problem-response pairs drawn from post-training, we identify two fundamental limitations of existing automated optimization modeling approaches: (L1) the sparsity of error-specific problems and (L2) the sparse rewards associated with difficult problems. We demonstrate that these limitations can result in suboptimal performance in domain-specific post-training for LLMs. To tackle the above two limitations, we propose a novel error-driven learning framework -- namely, auto\textbf{m}ated opt\textbf{i}mization modeli\textbf{n}g via a localizable error-\textbf{d}riven perspective (MIND) -- that customizes the whole model training framework from data synthesis to post-training. MIND is based on our key observation of the unique localizable patterns in error propagation of optimization modelings, that is, modeling errors may remain localized to specific semantic segments and do not propagate throughout the entire solution. Thus, in contrast to holistic reasoning tasks such as mathematical proofs, MIND leverages the construction of a focused, high-density training corpus and proposes \textbf{D}ynamic Supervised \textbf{F}ine-Tuning \textbf{P}olicy \textbf{O}ptimization (DFPO) to tackle difficult problems through localized refinement. Experiments on six benchmarks demonstrate that MIND consistently outperforms all the state-of-the-art automated optimization modeling approaches.

1 INTRODUCTION

Automated optimization modeling translates real-world problems into mathematical and executable representations, but existing post-training approaches face sparse error-specific data and limited learning signals on difficult problems. MIND addresses these challenges through error-driven data synthesis and dynamic post-training, and is evaluated across six benchmarks.

Motivation: Optimization modeling formalizes real-world problems by defining variables, objectives, and constraints for intelligent decision-making.The supplied passage introduces optimization as a tool across engineering, economics, logistics, manufacturing, and artificial intelligence.
Prior approaches: Existing automated optimization modeling methods use synthesized data followed by fine-tuning or adapt reinforcement learning techniques to the domain.Examples include ORLM, ReSocratic, Step-Opt, OptMATH, LLMOPT, and SIRL.
Challenges: MIND targets the sparsity of error-specific problems and the scarcity of learning signals on difficult problems.These are identified as two challenges in automated optimization modeling.
MIND: MIND combines reverse error-driven data synthesis with Dynamic Supervised Fine-Tuning Policy Optimization to customize training from data construction through post-training.The framework constructs MIND-Train and uses DFPO to dynamically correct wrong responses while keeping corrected responses close to the base model distribution.
Evaluation: Experiments on six benchmarks show that MIND outperforms state-of-the-art automated optimization modeling methods.The paper also introduces the MIND-Train dataset and MIND-Bench benchmark.

2 RELATED WORK

Prior work applies data synthesis, augmentation, fine-tuning, alignment, and reinforcement learning to improve domain-specific language-model capabilities. Automated optimization modeling studies have generally adapted these techniques without fully accounting for the domain’s distinctive characteristics.

Domain-specific Data Synthesis and Augmentation: Data generation research distinguishes augmentation of existing samples from synthesis of entirely new samples.Augmentation includes labeling, reformation, and co-annotation, while synthesis can generate samples from scratch or with generative models.
Domain-specific Post-Training: Domain-specific post-training includes fine-tuning, alignment, and reasoning methods, with RLVR benefiting tasks that have verifiable answers.The passage identifies mathematics and code generation as examples where verifiable rewards have supported progress.
Domain-specific Post-Training: Automated optimization modeling studies have applied general post-training techniques but overlooked characteristics unique to the domain.The cited approaches include supervised fine-tuning and reinforcement-learning-based methods such as LLMOPT and SIRL.

3 PRELIMINARIES

Automated optimization modeling generates a reasoning path that connects a natural-language problem to a mathematical formulation and executable code whose solver output should match the ground-truth objective. Preliminary analysis finds that errors are often localized to formulation components, motivating targeted data synthesis.

3.1 AUTOMATED OPTIMIZATION MODELING: An automated optimization modeling instance consists of a question, reasoning path, and resulting objective value.The reasoning path contains a rationale, mathematical formulation, and executable code.
3.1 AUTOMATED OPTIMIZATION MODELING: The task transforms a natural-language question into executable optimization code whose solver-computed objective matches the ground-truth objective.The backbone solver extracts code from the reasoning path, computes an objective value, and compares it with the ground truth to produce a reward.
3.2 PRELIMINARY RESULTS: Optimization modeling formulations typically contain variables, constraints, and objectives, which provide the components used for error analysis.The preliminary experiment compares generated code with ground-truth formulations using an LLM-as-a-judge approach.
3.2 PRELIMINARY RESULTS: 0.33 average error ratio indicates that incorrect formulations usually contain only a small fraction of errors rather than being entirely wrong.The passage links this pattern to relatively independent variables, constraints, and objectives and limited error propagation.
3.2 PRELIMINARY RESULTS: Different formulation components exhibit different recurring error types, motivating synthesis that explicitly incorporates common error patterns.The passage gives variable data-type selection, such as integer versus continuous, as one example.

4 METHODOLOGY

MIND builds challenging optimization-modeling data from localized error patterns, then combines fidelity-aware rewards with teacher-guided supervised refinement for difficult problems.

Error-Driven Reverse Data Synthesis: The reverse synthesis pipeline uses existing optimization-modeling datasets as seeds and deliberately targets common LLM error patterns.This avoids costly high-quality seed-data collection while producing more challenging training examples.
Error-Driven Reverse Data Synthesis: Single-error synthesis embeds one specified error pattern, whereas multi-error synthesis creates problems with multiple potential error-prone points.The two strategies address questions containing either one or several error types.
Error-Driven Reverse Data Synthesis: Two-stage quality control retains generated instances only after executable-solution validation and bidirectional problem–solution validation.The first stage checks solver execution and reasonable solutions; the second checks consistency between evolved problems and solutions.
Illustrative Error Patterns: The funding example exposes integer-variable definitions and incorrectly scaled objective coefficients as modeling errors.The intended funding amounts may be fractional, and the objective coefficients should use 0.2 and 0.25 rather than 200 and 250 million-dollar units.
Error-Driven Reverse Data Synthesis: Reverse synthesis can combine error patterns from different training datasets or industry scenarios, reducing reliance on costly expert annotations.This supports more diverse and challenging data generation with improved scalability and practicality.
Error-Driven Post-Training: DFPO addresses sparse rewards by using a stronger teacher LLM to refine incorrect responses and jointly optimizing reinforcement-learning and supervised-fine-tuning losses.The design targets insufficient guidance and distributional shifting in prior difficult-sample methods.
Error-Driven Post-Training: MIND’s reward combines modeling fidelity and answer accuracy, with fidelity based on normalized objective deviation and accuracy awarding 1 for a correct answer and 0 otherwise.The fidelity component supplies partial credit when a formulation is close to, but not identical with, the ground truth.

5 EXPERIMENTS

The experiments evaluate MIND across six benchmarks, compare it with automated modeling baselines, and ablate its data synthesis and DFPO post-training components. MIND improves base-model performance, with larger gains on challenging benchmarks and competitive generalization to MIND-Bench.

Main results: MIND-Qwen2.5-7B improves base-model performance by approximately 14.3% across six benchmarks, while MIND-Qwen3-8B improves its base model by approximately 31.0%.Gains average 24.1% on the more challenging IndustryOR, ComplexLP, and OptMATH benchmarks, but are marginal on tabular-data-heavy OptiBench.
Main results: MIND-Qwen2.5-7B achieves superior average performance to comparable-size baselines, while MIND-Qwen3-8B remains competitive with larger models.The comparison includes proprietary, agent-based, and training-based approaches.
Data synthesis ablation: Training on MIND-3K produces consistently higher accuracy gains during training and outperforms OR-Instruct-Data-3K on most benchmarks.This ablation indicates higher sample efficiency for error-driven reverse data synthesis.
Post-training ablation: DFPO outperforms DAPO by about 1.9% in macro-average accuracy across six benchmarks and by 10.2% on OptMATH.The comparison uses the same 10,000-instance dataset and reward design; DFPO is intended to provide additional learning signals on difficult problems.
Generalization study: On MIND-Bench, MIND-Qwen2.5-7B generalizes better than SIRL-Qwen2.5-7B, while MIND-Qwen3-8B is competitive with several larger models.MIND-Qwen2.5-7B still trails the 671B-parameter Deepseek-V3 and Deepseek-R1 models.

6 CONCLUSION

The paper concludes that optimization-modeling errors often remain localized within semantic segments, motivating an error-driven framework that customizes data synthesis and post-training. It also situates the evaluation across diverse optimization benchmarks and training resources.

Conclusion: Modeling errors are often localized within specific semantic segments, rather than propagating throughout the entire solution.This observation motivates the paper’s error-driven learning framework.
Conclusion: MIND customizes the model-training framework from data synthesis through post-training to address localized modeling errors and difficult problems with sparse reinforcement-learning signals.The paper identifies data diversity, quality, and quantity as important for domain-specific performance.
Evaluation scope: The evaluation covers benchmarks spanning linear, mixed-integer, integer, nonlinear, and second-order cone programming, including industrial, textbook, and long-context settings.The supplied benchmark descriptions include NL4Opt, MAMO, IndustryOR, OptiBench, and OptMATH-Bench.
Training resources: The training resources include OR-Instruct-Data-3K with 3,000 instances and OptMATH-Train with 200,000 instances, each containing questions, formulations, and code.These datasets are described as released by ORLM and OptMATH, respectively.

A.3 MIND-TRAIN DATASET

MIND-Train is documented through dataset statistics, topic analyses, length distributions, and examples of single-error and multi-error reverse synthesis. Its 10,000 training instances combine MIND-Train with two existing datasets.

Dataset construction: MIND-Train statistics summarize question examples across three stages of the reverse data synthesis pipeline.The appendix also presents a multi-error example alongside a single-error example.
Dataset characteristics: The training corpus covers diverse optimization topics, including hospitals, transportation, machines, warehouses, surgery, facilities, energy, and products.The topic distribution is summarized through a word cloud.
Dataset characteristics: The top 50 gerund pairs represent typical optimization-modeling patterns in MIND-Train.These pairs are extracted from the training questions.
Length distribution: Prompts average 392 words and responses average 790 words across the 10,000-instance training dataset.Most prompts fall between 200–600 words, while most responses fall between 500 and 1,200 words.
Dataset construction: The final training mixture contains 5,000 MIND-Train instances, 1,000 OR-Instruct-Data-3K instances, and 4,000 OptMATH-Train instances.Together, these sources provide 10,000 instances for Qwen2.5-7B-Instruct training.

A.4 MIND-BENCH DATASET

MIND-Bench contains 69 out-of-distribution operations-research questions from textbooks and industry scenarios, with no ambiguity about variable types. The appendix also specifies prompts and scoring procedures for generating, correcting, and evaluating mathematical models and code.

Benchmark construction: MIND-Bench contains 69 questions derived from textbooks and industry scenarios that differ from the sources of other public benchmarks and training datasets.The benchmark examples are shown in Figure 12, with scenario statistics in Figure 11.
Benchmark construction: MIND-Bench questions have unambiguous variable types, so the benchmark does not use the rule-based substitution method applied elsewhere.This design isolates modeling performance without ambiguity about integer versus continuous variables.
Error evaluation: The preliminary evaluation prompt computes formulation size from variables, objectives, and constraints, then compares incorrect or missing components with the correct formulation.The task is to assess consistency between the correct formulation and its PySCIPOpt implementation.
Data synthesis prompts: The synthesis prompts compare two optimization-modeling problems to identify discrepancies such as missing constraints, incorrect coefficients, wrong variable types, and incorrect objective direction.The multi-error template then constructs more challenging instances containing multiple error-prone points.
Modeling prompts: The standard modeling prompt asks the model to analyze the problem, define sets, parameters, variables, objectives, and constraints, and provide PySCIPOpt code.The pipeline separates reasoning, mathematical formulation, and code generation stages.
Correction prompts: Correction prompts use a correct response as reference, require reasoning about changes, and output the entire corrected response while preserving already-correct content.They instruct the model to change only incorrect portions and retain the original coding style.

C.2 PRELIMINARY RESULTS ON DEEPSEEK-V3

Deepseek-V3 errors are generally confined to a small fraction of each formulation rather than making the entire formulation incorrect, supporting localized error propagation.

Deepseek-V3 usually introduces only a small fraction of errors when it produces an incorrect formulation.The analysis uses 100 incorrect generation results to examine this pattern.
The observed error distribution further supports the conclusion that errors remain localized rather than propagating throughout the formulation.
These findings extend the preliminary error-propagation analysis beyond Qwen2.5-7B to a model with a different architecture.

C.3 REWARD WEIGHT SENSITIVITY ANALYSIS

The reward-weight analysis finds that moderate fidelity-reward weights perform best on most benchmarks, while excessive weighting is less effective.

The sensitivity study evaluates four reward-function settings: α ∈ {0.0, 0.2, 0.4, 0.6}.
α = 0.2 and α = 0.4 achieve better performance on most benchmarks than α = 0.0 and α = 0.6.The experiments test α values of 0.0, 0.2, 0.4, and 0.6 using DAPO on 10,000 instances for seven epochs.
The fidelity reward should serve as an auxiliary signal rather than dominate the final reward value.Here, α = 0.0 corresponds to a standard 0-1 reward.

C.4 ABLATION STUDY OF DATA SYNTHESIS STRATEGIES

Mixed single-error and multi-error synthesis outperforms either strategy alone, while single-error data is more effective than multi-error data for training.

MIND-Mix-3K achieves better training performance than either MIND-Single-1.5K or MIND-Multi-1.5K.The mixed dataset combines 1,500 single-error and 1,500 multi-error instances.
MIND-Single-1.5K yields better performance than MIND-Multi-1.5K after seven training epochs across six benchmarks.
52.9% average accuracy on MIND-Single-1.5K exceeds the 41.2% achieved on MIND-Multi-1.5K.The reported gap supports the authors’ hypothesis that highly challenging datasets are harder for LLMs to learn from directly.

C.5 MODELING ERROR ANALYSIS

The error analysis shows that post-training shifts failures from basic structural mistakes toward more sophisticated modeling and scoping challenges, while case studies illustrate localized correction.

Error taxonomy: The five most frequent pre-training errors include incorrect objective terms (12.8%), incorrect decision variables (12.1%), incorrect constraints (11.8%), incorrect variable types (11.8%), and constraint omission (10.1%).
Error taxonomy: After DFPO, the leading errors are incorrect decision variables (15.5%), incorrect constraints (15.5%), advanced-technique errors (11.7%), decision-variable omission (10.6%), and constraint omission (8.5%).
Error progression: Basic syntactic or structural errors diminish after post-training, while advanced modeling and comprehensive problem-scoping errors become dominant.Examples include piecewise linearization, indicator constraints, and omitted variables or constraints.
Case studies: The case studies use examples from MAMO ComplexLP and OptMATH to illustrate corrected wrong responses.
Case studies: In the diet example, an incorrect variable type leaves the parameters, objective function, and constraints correct, so correcting that local error fixes the answer.
Case studies: In the scheduling example, mistakes in machine-capacity and makespan constraints do not affect the correctness of the other constraint groups.Fixing only these localizable errors corrects the overall solution.