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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:2602.11164v1cs.LGcs.AIcs.CL

TL;DR

Automated optimization modeling post-training is constrained by sparse error-specific data and limited learning signals for difficult problems. MIND combines focused error-driven data synthesis with DFPO, and it outperforms state-of-the-art methods across six benchmarks.

  • Problem

    Automated optimization modeling post-training faces sparse error-specific problems and scarce learning signals for difficult problems.

    Source: §1 · p. 3

  • Method

    MIND combines error-driven reverse data synthesis for a focused training corpus with DFPO for localized refinement of difficult problems.

    Source: §1 · pp. 2–3

  • Results

    Across six benchmarks, MIND outperforms state-of-the-art automated optimization modeling methods.

    Source: §1 · p. 3, §5.2 · p. 9

  • Takeaways & Limitations

    The study advances understanding of LLMs for automated optimization modeling by highlighting the importance of domain-specific data and learning signals for difficult problems.

    Source: §6 · p. 11

    The reward design relies on the stated assumption that modeling-error degree can guide solutions toward structurally closer formulations.

    Source: §4.2 · p. 7

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

MIND addresses costly, sparse training data and sparse rewards by exploiting the localization of optimization-modeling errors in data synthesis and post-training. It outperforms state-of-the-art methods on six benchmarks and introduces a training dataset and benchmark.

Background

  • LLMs can translate natural-language problem descriptions into mathematical formulations and solver code, rapidly generating candidate formulations that support human experts despite not guaranteeing complete accuracy.Optimization models define decision variables, objectives, and constraints so solvers can compute solutions.Source: §1 · pp. 1–2

Motivation and contributions

  • The paper identifies costly high-quality data generation, limited generalization beyond seed data, and sparse final-outcome rewards on difficult problems as key challenges.Its analysis finds a low error ratio and errors localized to subsets such as variables, constraints, or objectives, motivating focused data generation and training.Source: §1 · p. 2

MIND framework

  • MIND combines error-driven synthesis of a focused, high-density training corpus with DFPO, which corrects wrong responses while keeping generated corrections close to the base model’s response distribution.DFPO integrates supervised fine-tuning and reinforcement learning by leveraging this slight distributional discrepancy.Source: §1 · p. 2

Contributions

  • MIND outperforms state-of-the-art automated optimization modeling methods on six benchmarks and releases the MIND-Train dataset and MIND-Bench benchmark.The paper presents these resources for the automated optimization research community.Source: §1 · p. 3

2 RELATED WORK

Prior work adapts general data synthesis and post-training techniques to automated optimization modeling, but overlooks the domain’s distinctive characteristics, motivating MIND’s error-driven approach.

Domain-specific Data Synthesis and Augmentation

  • Prior optimization-modeling data pipelines use forward augmentation and synthesis (ORLM), iterative expansion (Step-Opt (Wu et al., 2025)), reverse synthesis (Resocratic (Yang et al., 2024)), or bidirectional checks (OptMATH (Lu et al., 2025)), yet overlook domain-specific data characteristics.OptMATH compares formulations generated in both directions to check data quality.Source: §2 · p. 3

Domain-specific Post-Training

  • General post-training spans fine-tuning, alignment, and reasoning; verifiable math and code have advanced RLVR, with GRPO replacing PPO’s value function with within-group response-level advantages (Shao et al., 2024).DAPO (Yu et al., 2025) adds decoupled clipping, dynamic filtering, token-level loss aggregation, and reward shaping; GHPO (Liu et al., 2025) uses solution hints, while VAPO (Yue et al., 2025) uses a value model and adds negative log-likelihood loss for correct samples.Source: §2 · p. 3
  • In optimization modeling, ORLM, Step-Opt, Resocratic, and OptMATH study SFT, LLMOPT (Jiang et al., 2024) explores Kahneman–Tversky Optimization, and SIRL (Chen et al., 2025) examines RLVR, but these approaches overlook domain-specific characteristics.The authors argue that RLVR can bridge general-purpose LLMs and automated optimization modeling when guided by an error-driven perspective.Source: §2 · pp. 3–4

3 PRELIMINARIES

Automated optimization modeling transforms a natural-language question into a rationale, mathematical formulation, and executable code whose solver-computed objective matches the ground truth. Preliminary results suggest modeling errors are often localized to a small fraction of a formulation, motivating error-focused data synthesis.

3.1 AUTOMATED OPTIMIZATION MODELING

  • An instance maps question q to reasoning path o and objective a; correct modeling requires solver-executable code in o to produce the ground-truth objective a∗.The reasoning path comprises rationale Z, mathematical formulation MF, and executable code C.Source: §3.1 · p. 4
  • The policy generates reasoning path o from q, a backbone solver extracts code C and computes a, and the reward compares a with ground truth a∗.The training setup is described using dataset D, policy parameters θ, and a backbone solver such as PySCIPOpt.Source: §3.1 · p. 4

3.2 PRELIMINARY RESULTS

  • Preliminary error profiling used Qwen-2.5-7B-Instruct on ORLM and an LLM judge to compare generated code with ground-truth formulations across variables, constraints, and objectives.Observed component-specific error patterns motivated collecting frequent errors and synthesizing data that explicitly incorporates them.Source: §3.2 · pp. 4–5
  • An average error ratio of 0.33 across 100 incorrect generations suggests errors affect a small fraction of formulations and are relatively independent across variables, constraints, and objectives.Figure 2 shows the error-ratio distribution; errors were assessed by comparing generated code with ground-truth formulations.Source: §3.2 · p. 4, Figure 2 · p. 4

4 METHODOLOGY

MIND combines error-driven reverse data synthesis with DFPO: the synthesis pipeline creates validated, error-targeted training examples, while DFPO uses fidelity rewards and teacher-refined responses to guide post-training.

4.1 MIND: ERROR-DRIVEN REVERSE DATA SYNTHESIS PIPELINE

  • MIND seeds reverse synthesis from existing optimization datasets and targets common LLM error patterns, avoiding costly high-quality seed-data collection and reducing reliance on expert annotations.The pipeline is designed to support diverse, challenging data from different training datasets or industry scenarios.Source: §4.1 · pp. 5–6
  • The pipeline identifies errors by comparing base-model-generated code with ground-truth formulations on seed problems, using powerful LLMs to extract error patterns.Seeds come from OR-Instruct-Data-3K and OptMATH-Train.Source: §4.1 · p. 5
  • Reverse synthesis embeds either one targeted error trap or multiple potential error-prone points in new problems, and generates a corresponding modeling solution for each.The single-error and multi-error strategies address questions that may contain multiple error types.Source: §4.1 · p. 5
  • Two-stage quality control retains examples only when solver validation yields an executable, reasonable solution and an independent LLM’s solution matches the synthesized ground-truth objective value.The pipeline’s stages are error-pattern identification, reverse synthesis, and quality control.Source: §4.1 · p. 5

4.2 MIND: ERROR-DRIVEN POST-TRAINING METHOD

  • A fidelity reward gives partial credit for formulations close to ground truth, based on the hypothesis that modeling error and objective deviation are positively correlated in expectation.Modeling fidelity measures formulation discrepancies; the reward is intended to guide outputs toward structurally closer formulations.Source: §4.2 · pp. 6–7
  • DFPO refines incorrect rollouts with a stronger teacher using ground-truth solutions, then combines SFT loss with RL loss to address sparse rewards and distributional shift.The teacher’s corrections are designed to remain accurate and close to the base model’s response distribution.Source: §4.2 · p. 7

5 EXPERIMENTS

Experiments show that MIND improves automated optimization modeling across benchmarks and base models, with gains especially pronounced on challenging tasks. Ablations attribute benefits to error-driven data synthesis and DFPO, and a new benchmark indicates generalization beyond the training setting.

5.1 EXPERIMENTAL SETUP

  • The experiments use Qwen-2.5-7B-Instruct and Qwen3-8B, evaluate six benchmarks against LLM and optimization-modeling baselines, and report zero-shot pass@1 accuracy.Correctness requires the relative error between predicted and ground-truth objective values to be below 10^-6; training uses 10,000 sampled instances synthesized from OR-Instruct-Data-3K and OptMATH-Train.Source: §5.1 · pp. 8–9

5.2 MAIN RESULTS

  • MIND-Qwen2.5-7B improves its base model by approximately 14.3% across six benchmarks, with an average 24.1% gain on challenging benchmarks.MIND-Qwen3-8B improves its base model by approximately 31.0%; gains on tabular-data-heavy OptiBench are marginal, possibly because similar problem types are limited in training data.Source: §5.2 · p. 9, Table 1 · p. 9
  • MIND-Qwen2.5-7B has superior average performance to comparable-size baselines, while MIND-Qwen3-8B is competitive with larger models including GPT-4 and Deepseek-V3.The reported comparisons cover proprietary, agent-based, and training-based approaches; MIND-Qwen2.5-7B particularly improves on prior training-based methods on IndustryOR, ComplexLP, and OptMATH.Source: §5.2 · p. 9, Table 1 · p. 9

5.3 ABLATION STUDY

  • Training on 3,000-instance MIND-3K yields higher accuracy gains as training progresses and outperforms OR-Instruct-Data-3K on most benchmarks after seven epochs.This comparison uses DAPO to train Qwen-2.5-7B-Instruct on each dataset and supports greater sample efficiency from reverse data synthesis.Source: §5.3 · pp. 9–10, Figure 6 · p. 10, Table 2 · p. 10
  • DFPO exceeds DAPO by about 1.9% in six-benchmark macro-average accuracy and by 10.2% on OptMATH, providing stronger learning signals for difficult problems.The comparison controls for the 10,000-instance dataset and chain-of-thought prompt; SFT alone gives no significant gains, while SFT warm-started GRPO improves but still trails DFPO on challenging benchmarks.Source: §5.3 · p. 10, Table 3 · p. 10

5.4 GENERALIZATION STUDY (RQ5)

  • On the 69-problem MIND-Bench, MIND-Qwen2.5-7B generalizes better than SIRL-Qwen2.5-7B but trails 671B-parameter Deepseek-V3 and Deepseek-R1.MIND-Qwen3-8B is competitive with Deepseek-V3, Deepseek-R1, and SIRL-Qwen2.5-32B.Source: §5.4 · p. 10, Table 4 · p. 10

6 CONCLUSION … A.2 SEED DATASET

The conclusion presents MIND as an error-driven framework motivated by localized modeling errors and data and post-training limitations. The appendix describes benchmark datasets, evaluation criteria, and seed training corpora.

6 CONCLUSION

  • MIND builds on the finding that modeling errors often remain localized and customizes training from data synthesis through post-training.The paper highlights the need for diverse, high-quality, sufficient training data and notes that reinforcement learning alone can provide too few learning signals on difficult problems.Source: §6 · p. 11

A.1 BENCHMARK DATASET

  • NL4OPT offers 245 validated, high-quality LP questions and is described as an easy benchmark with simple constraints and scenarios (Ramamonjison et al., 2023; Lu et al., 2025).It is identified as the first curated dataset derived from the NL4OPT Competition.Source: §A.1 · p. 15
  • MAMO contains 642 EasyLP and 203 ComplexLP questions, covering LP and MILP with a primary emphasis on MILP (Chen et al., 2025).The passage notes that MILP constitutes the majority of real-world optimization problems.Source: §A.1 · p. 15
  • IndustryOR comprises 100 questions from real-world scenarios across sectors and targets industrial complexity with IP, LP, MILP, NLP, and specialized formulations (Chen et al., 2025).Source: §A.1 · p. 15
  • OptiBench contains 605 textbook-derived questions spanning IP, LP, MILP, and NLP, with extensive tabular data for testing table understanding and reasoning (Yang et al., 2024).The cited textbooks are Bertsimas & Tsitsiklis (1997), Conforti et al. (2014), and Wolsey (2020).Source: §A.1 · pp. 15–16
  • OptMATH-Bench contains 166 expert-curated questions across IP, LP, MILP, NLP, and SOCP, testing long-context modeling with longer contexts and more complex constraints (Lu et al., 2025).Source: §A.1 · p. 16
  • For ambiguous variable types, evaluation accepts either integer or continuous assumptions when the optimal solution matches ground truth, defined by an objective absolute difference below 10^-6.The approach follows OptMATH (Lu et al., 2025) and SIRL (Chen et al., 2025), using rule-based substitution.Source: §A.1 · p. 16

A.2 SEED DATASET

  • The seed corpora are OR-Instruct-Data-3K, with 3,000 instances from a 30,000-example ORLM set, and OptMATH-Train, with 200,000 instances; both include questions, formulations, and code.Source: §A.2 · p. 16

A.3 MIND-TRAIN DATASET

MIND-Train is profiled through its reverse-synthesis stages, topic and modeling-pattern analyses, length distributions, and final training-set composition.

MIND-Train Statistics

  • MIND-Train statistics summarize question examples across three reverse-synthesis stages; the single-error strategy uses DeepSeek-R1-0528 and the multi-error strategy uses DeepSeek-V3.1-Think.The paper also presents a multi-error example and notes that error pattern 1 comes from Figure 7, while error pattern 2 comes from Figure 4.Source: §A.3 · p. 16, Table 5 · p. 16, Figure 7 · p. 17

Word Cloud Analysis

  • The word cloud highlights diverse optimization-modeling topics, including hospitals, transportation, machines, warehouses, surgery, facilities, energy, and products.Source: §A.3 · p. 16, Figure 8 · p. 17

Gerund Pairs Analysis

  • The 50 most frequent gerund pairs, extracted with en core web sm, represent typical optimization-modeling patterns in MIND-Train questions.Source: §A.3 · p. 16, Figure 9 · p. 18

Length Distribution Analysis

  • Prompts average 392 words, mostly 200–600, while responses average 790 words, mostly 500–1,200.Source: §A.3 · pp. 16, 18

Training-Set Composition

  • The 10,000-instance training set combines 5,000 MIND-Train, 1,000 OR-Instruct-Data-3K, and 4,000 OptMATH-Train examples for Qwen2.5-7B-Instruct.Source: §A.3 · p. 18

A.4 MIND-BENCH DATASET … C.1 TRAINING AND INFERENCE DETAILS

The appendices specify an out-of-distribution benchmark, error-driven and chain-of-thought prompt designs, dynamic-SFT correction procedures, an error-ratio assessment, and training and inference settings. Together, these details describe the data and implementation used to evaluate and train the modeling approach.

A.4 MIND-BENCH DATASET

  • MIND-Bench contains 69 textbook- or industry-derived questions from sources outside public benchmarks and training data, with unambiguous variable types and no rule-based substitution.Figure 11 reports scenario statistics; Figure 12 shows example problems.Source: §A.4 · p. 18, Figure 11 · p. 19, Figure 12 · p. 20

B.1 PROMPT TEMPLATE FOR PRELIMINARY RESULTS

  • The preliminary-results prompt measures formulation error by sizing the correct model and summing the sizes of logically incorrect or missing components in the implementation.It directs the model to report the resulting error ratio.Source: §B.1 · p. 21

B.2 PROMPT TEMPLATES FOR DATA SYNTHESIS

  • Single-error synthesis identifies erroneous and correct code portions, corrects the misapplied modeling pattern, then generates distinct, more complex instances reusing that pattern.The prompt asks for new problems whose other formulation and implementation components differ from the original; Deepseek-R1 is used for the reverse synthesis pipeline.Source: §B.2 · pp. 21–22
  • Multi-error synthesis combines the corrected patterns from two problems in each new, more complex instance, while requiring the rest of the formulation to differ substantially from both originals.The prompt directs the model to detect discrepancies, label erroneous and correct code, and derive each corrected pattern before combining them.Source: §B.2 · p. 23

B.3 PROMPT TEMPLATE FOR CHAIN-OF-THOUGHT

  • The chain-of-thought prompt stages modeling as problem analysis and rationale extraction, mathematical formulation, and translation into executable PySCIPOpt code.The paper reports using this sequence following DeepSeek-R1-Zero and SIRL.Source: §B.3 · p. 24

B.4 PROMPT TEMPLATE FOR DYNAMIC SFT

  • Dynamic-SFT prompts generate independent correct responses using ground-truth solutions as guidance, then correct rollout responses while preserving correct portions and changing only errors.The correction prompt asks for reasoning about modifications and the entire corrected response, without added commentary.Source: §B.4 · pp. 25–27

C.1 TRAINING AND INFERENCE DETAILS

  • Training ran on one node with four 80-GB NVIDIA A100 GPUs; ms-swift implemented SFT, while VeRL implemented GRPO, DAPO, and DFPO.Training hyperparameters are listed in Tables 6–9.Source: §C.1 · p. 27
  • Inference uses greedy decoding to ensure reproducibility, with hyperparameters listed in Table 10.Source: §C.1 · p. 27

C.2 PRELIMINARY RESULTS ON DEEPSEEK-V3

Preliminary experiments on DeepSeek-V3 support the finding that modeling errors often remain localized, and report a lower average error ratio than for Qwen2.5-7B-Instruct.

C.2 PRELIMINARY RESULTS ON DEEPSEEK-V3

  • On OR-Instruct-3K, DeepSeek-V3 has a lower average error ratio (29%) than Qwen2.5-7B-Instruct (33%) in preliminary experiments using a model with a different architecture.When errors occur, DeepSeek-V3 introduces only a small fraction of errors rather than entirely incorrect formulations in most cases.Source: §C.2 · p. 27

C.3 REWARD WEIGHT SENSITIVITY ANALYSIS

The reward-weight sensitivity analysis finds that intermediate fidelity-reward weights perform better on most benchmarks than either endpoint, suggesting the auxiliary fidelity reward should not dominate the final reward.

C.3 REWARD WEIGHT SENSITIVITY ANALYSIS

  • The analysis tests α values of 0.0, 0.2, 0.4, and 0.6 using DAPO to train Qwen2.5-7B-Instruct on 10,000 instances for 7 epochs.Source: §C.3 · p. 27
  • α = 0.2 and α = 0.4 achieve better performance on most benchmarks than α = 0.0 and α = 0.6, so fidelity reward should not dominate the final reward.α = 0.0 corresponds to a standard 0-1 reward.Source: §C.3 · pp. 27, 30

C.4 ABLATION STUDY OF DATA SYNTHESIS STRATEGIES

MIND-Mix-3K yields better training performance than either single-error or multi-error data, while single-error training outperforms multi-error training. The authors attribute the latter gap to the greater difficulty of multi-error datasets.

C.4 ABLATION STUDY OF DATA SYNTHESIS STRATEGIES

  • 52.9% average accuracy on MIND-Single-1.5K versus 41.2% on MIND-Multi-1.5K supports the claim that multi-error synthesis produces substantially more difficult data.The authors hypothesize that LLMs struggle to learn effectively when trained directly on highly challenging datasets.Source: §C.4 · p. 30
  • The ablation split MIND-3K into two 1,500-instance datasets and trained Qwen2.5-7B-Instruct from scratch on each using DAPO.The compared datasets were MIND-Single-1.5K and MIND-Multi-1.5K.Source: §C.4 · p. 30
  • MIND-Mix-3K achieves better training performance than MIND-Single-1.5K and MIND-Multi-1.5K, and after seven epochs MIND-Single-1.5K outperforms MIND-Multi-1.5K across six benchmarks.Table 11 presents the detailed benchmark comparison; the passage does not provide its individual metric values.Source: Table 11 · p. 30, §C.4 · p. 30

C.5 MODELING ERROR ANALYSIS · C.6 CASE STUDY: EXAMPLES OF CORRECTED WRONG RESPONSES FOR DYNAMIC SFT

DFPO shifts dominant modeling failures from basic errors toward harder semantic challenges, while case studies show that correcting localized errors can repair an answer without revising its unaffected parts.

C.5 MODELING ERROR ANALYSIS

  • The analysis samples 300 erroneous responses from each model and uses three independent domain-expert annotations per query-response pair, with high inter-annotator agreement.The models are Qwen2.5-7B-Instruct before post-training and MIND-Qwen2.5-7B after DFPO-based post-training.Source: §C.5 · pp. 30, 32
  • The most frequent errors change from incorrect objective terms (12.8%) before post-training to incorrect decision variables and incorrect constraints (15.5% each) after DFPO.Before post-training, the other top errors are incorrect decision variables (12.1%), incorrect constraint (11.8%), incorrect variable types (11.8%), and constraint omission (10.1%). After DFPO, the remaining top errors are incorrect or missing advanced modeling techniques (11.7%), decision variable omission (10.6%), and constraint omission (8.5%).Source: Table 12 · p. 32, §C.5 · p. 32
  • After DFPO, dominant failures shift from surface-level mistakes toward advanced modeling and problem-scoping challenges.The paper characterizes this shift as evidence that DFPO mitigates simpler errors while leaving deeper semantic challenges as prominent failure modes.Source: §C.5 · pp. 32–33

C.6 CASE STUDY: EXAMPLES OF CORRECTED WRONG RESPONSES FOR DYNAMIC SFT

  • Examples from MAMO ComplexLP and OptMATH show a powerful LLM correcting base-model errors while producing responses whose distribution closely matches the wrong responses.The examples are shown in Figures 18 and 19, respectively.Source: §C.6 · p. 33, Figure 18 · p. 34, Figure 19 · p. 35
  • In the diet example, food quantities should be continuous because the problem does not require whole units, and the variable-type error leaves the parameters, objective, and constraints correct.This illustrates that one localized modeling error need not invalidate other solution components.Source: §C.6 · p. 33, Figure 16 · p. 33, Figure 18 · p. 34
  • Together, the cases support the paper’s conclusion that modeling errors can remain localized, allowing an overall answer to be corrected by fixing only those errors.The scheduling example specifically shows that errors in makespan constraints do not compromise precedence, machine-capacity, or non-negativity constraints, and vice versa for machine-capacity errors.Source: §C.6 · p. 36
  • In the scheduling example, minor machine-capacity and makespan errors do not affect the other listed constraint groups, so correcting those errors can repair the solution.The unaffected groups include precedence and non-negativity constraints, as well as the other of the two error-bearing constraint groups.Source: Figure 17 · p. 33, Figure 19 · p. 35, §C.6 · p. 36

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