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DualSchool: How Reliable are LLMs for Optimization Education?

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Open-weight LLMs can recite the primal-to-dual recipe but fail to carry it out; the best test score is 47.8%.

desk verdict DualSchool is a solid benchmark with a genuinely useful canonicalization method, but the headline accuracies rest on primals generated and validated by the very objective-value check the paper criticizes. read the letter →

arxiv 2505.21775 v1 pith:Y2JXIK3I submitted 2025-05-27 cs.LG cs.AImath.OC

Michael Klamkin , Arnaud Deza , Sikai Cheng , Haoruo Zhao , Pascal Van Hentenryck This is my paper · ORCID
classification cs.LGcs.AImath.OC
keywords primal-dualconversionLLMreliabilitylinearprogrammingoptimizationeducationbenchmarkgrapheditdistancedualizationevaluation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces DualSchool, a benchmark and verification framework for primal-to-dual conversion (P2DC) in linear programming, along with derivative tasks in verification, correction, and error classification. It sets out to test a student-facing expectation: because LLMs have seen the conversion rules and many worked examples, they should handle this introductory optimization procedure reliably. The central finding is that they do not: the best open-weight model tested reaches only 47.8% accuracy on the NL4OPT subset, and even two-variable instances cause failures, despite the same models reciting the conversion procedure correctly when asked abstractly. The paper offers this gap as a caution for optimization education and as a structured testbed for reasoning systems.

What carries the argument

The central object is Canonical Graph Edit Distance (CGED), a graph-based equivalence checker that considers candidates correct only when the graph edit distance to the ground-truth dual is zero. Before comparing graphs, it canonicalizes two convention differences that arise across dualization procedures: slack variables and variable sign choices. That preprocessing distinguishes CGED from normalized graph edit distance, which would mark many valid convention variants as incorrect, and from objective-value matching, which can pass structurally wrong duals.

What would settle it

Re-run the generation evaluation on the published DualSchool dataset after replacing every language-model-generated primal that passed only the objective-value screen with a formulation whose correctness is verified independently, for example by exact reformulation and dual feasibility checks, and then compare CGED accuracies; if the 47.8% figure moves substantially, the original experiment was measuring label quality rather than dualization skill.

Watch

Extended reading notes

Core claim

The paper's core claim is that state-of-the-art open LLMs are unreliable at primal-to-dual conversion. Using DualSchool, which pairs each primal with a ground-truth dual generated by symbolic dualization and checks candidates with the Canonical Graph Edit Distance (CGED), the paper reports that the strongest model tested scores 47.8% on the NL4OPT instances, with most models far lower, and that accuracy does not improve systematically with one-shot prompting. The failures include duals with correct objective values but wrong structure, such as omitted variable bounds, so objective-value evaluation overstates competence. The same models can state the P2DC procedure correctly when asked as a recipe, showing a gap between recitation and execution; derived tasks such as correcting, verifying, and classifying duals also stay near or below random-guess baselines.

Load-bearing premise

The benchmark's ground-truth primal-dual pairs are correct; in particular, the primals produced by Llama 3.3 and accepted only when their objective value matched are treated as genuine linear programs.

Editorial extensions

If this is right

  • Students and instructors cannot assume an LLM's generated dual is correct, even when the formulation parses, runs, and gives the expected objective value.
  • Objective-value-based evaluation, the common practice in LLM4OPT, overstates model competence on structured optimization tasks; CGED offers a stricter signal.
  • The published dataset and automatic CGED checker let future work use P2DC as a fine-tuning or reinforcement-learning reward signal rather than a one-off quiz.
  • One-shot prompting does not reliably help, so simply adding an example to the prompt is unlikely to close the gap.
  • Because the derivative tasks stay near random-guess baselines, LLMs also cannot yet be trusted to verify or repair their own dualization output.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the ground-truth labels are audited and the model-generated primals that were filtered only by objective value turn out to be misformulated, part of the reported failure could be label noise; such an audit would separate benchmark-quality issues from model-capability issues.
  • CGED's canonicalization may generalize beyond P2DC as a normalizer for comparing linear programs written under different conventions, with the caveat that it deliberately forgets scaling and variable substitutions.
  • A natural testable extension is to see whether a reasoning-focused model trained with CGED as reward crosses a high-accuracy threshold on the same instances, which would indicate the failure is trainable rather than structural.
  • The strong bias toward answering 'no' in the verification task, if it holds outside this benchmark, could make LLM-based grading of student optimization work systematically harsh in one direction while missing real errors in the other.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper introduces DualSchool, a benchmark and evaluation framework for testing whether open-weight LLMs can perform primal-to-dual conversion (P2DC) of linear programs. The framework combines four tasks—generation, correction, verification, and error classification—and proposes a new evaluation metric, the Canonical Graph Edit Distance (CGED), which canonicalizes slack variables and variable-sign conventions before computing graph edit distance. The benchmark includes over 1,300 LP instances drawn from existing datasets and synthetic sources, with ground-truth duals computed symbolically using Dualization.jl. Experiments on several open LLMs (Mistral, Phi, Gemma, Qwen, Llama) report uniformly low accuracies: the best model, Phi-4, achieves only 47.8% CGED accuracy on NL4OPT instances, and performance on synthetic two-variable instances is even lower. The paper concludes that LLMs can recite dualization procedures but cannot reliably execute them, with implications for AI-assisted education.

Significance. If the results are trustworthy, DualSchool provides a valuable, reproducible benchmark for a well-defined structured task in optimization education and reasoning. The CGED metric is a thoughtful contribution that addresses a real gap: existing evaluation methods are either too permissive (objective-value checks) or too restrictive (raw graph edit distance) for judging correctness of duals, which can legitimately differ by convention. The paper ships open code and data, uses machine-checked symbolic dualization for ground truth, and reports results across multiple model families and prompting settings, which are concrete strengths. The finding that LLMs can recite the procedure yet fail on tiny instances is important and likely to influence future work on LLM reliability in OR. However, the benchmark's label generation for a large subset of instances uses the very method the paper criticizes, which undermines the quantitative claims until that issue is resolved.

major comments (2)
  1. [Appendix A.3, Section 4, Table 2] The ground-truth labels for the NL4OPT, EasyLP, and ComplexOR subsets (805 of 1,319 instances, Table 1) are derived from primal formulations generated by Llama 3.3 and accepted solely on the basis of matching the published objective value, retrying up to five times. Section 4 explicitly argues that objective-value checks admit false positives (e.g., omitted non-tight constraints) and are reward-hackable. Therefore, the generated primal may not be the intended LP, and the symbolic dual computed by Dualization.jl is then the dual of a possibly wrong problem. The reported accuracies for these subsets—including the headline 47.8% on NL4OPT—are not trustworthy as evidence about LLM performance on the intended P2DC task. The paper provides no human or independent verification of even a sample of these labels. This issue is load-bearing for the paper's central quantitative claim.
  2. [Section 5.3, Figures 2-3] The CORRECTION, VERIFICATION, and CLASSIFICATION tasks on the same three datasets inherit the same label corruption: the error-injected duals are derived from ground-truth duals of Llama-generated primals. If a primal is incorrect, then the 'ground truth' dual and the injected error types are not well-defined with respect to the original source problem, so the reported low accuracies on these derivative tasks may be depressed or otherwise distorted. The paper should either exclude these instances from the evaluation or provide strong evidence that the generated primals match the intended problems, such as a human-verified subsample or an alternative correctness check beyond objective value.
minor comments (4)
  1. [Section 5, Table 1] The word 'poltyopes' in the benchmark description is a typo for 'polytopes'.
  2. [Table 2] The table layout is difficult to read because each dataset's metrics are repeated without clear visual separation; a separate subtable per dataset or a grouped header would improve clarity.
  3. [Appendix B.1] The manual evaluation of 'LLMs know the procedure' lacks a detailed scoring rubric; the authors should state how they determined whether a response was a valid procedure.
  4. [Section 4.1] The description of the variable-sign canonicalization for double-sided bounded variables is somewhat dense; a short worked example with the actual coefficient flips would aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: DualSchool's evaluation compares LLM outputs against externally computed symbolic duals; no fitted parameter is reused as a prediction and no load-bearing self-citation appears.

full rationale

The paper's central claim is an empirical measurement: LLMs are prompted to produce gurobipy code for a dual, and the resulting MPS models are compared by canonical graph edit distance (CGED) to ground-truth duals obtained with Dualization.jl (Section 5, Appendix A.3). The ground-truth dual is not defined in terms of any LLM output, and CGED is a fixed, parameter-free comparison procedure rather than a fitted model, so the reported accuracies (e.g., 47.8% for Phi 4 on NL4OPT) are not equivalent by construction to any input. There are no self-citations by the present authors invoked as load-bearing evidence, and no uniqueness theorem or ansatz is imported from prior work to force the conclusion. The only notable validity concern is that some benchmark primals from NL4OPT/EasyLP/ComplexOR were generated by Llama 3.3 and accepted via objective-value checks (Appendix A.3), even though Section 4 argues objective-value checks admit false positives. That is a potential label-corruption or contamination issue for the benchmark, not a circular derivation: the comparison still goes through an external symbolic dualization step, and no quantity is both fitted and predicted. Hence no circular step meeting the required evidentiary standard can be identified.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the validity of automatic dualization, on the correctness of LLM-generated primals for derived datasets, on treating execution failures as incorrect, and on the soundness of the CGED canonicalization. No new physical or formal objects are introduced.

free parameters (1)
  • retry_limit = 5
    Section A.3: LLM-generated primal formulations are re-tried until objective-value match, and excluded after five failed retries; this hand-chosen threshold changes the composition and difficulty of the NL4OPT/EasyLP/ComplexOR subsets.
assumptions (5)
  • domain assumption Dualization.jl automatic dualization produces a correct canonical dual for every primal.
    Section 3 and Appendix A.3 use Dualization.jl to generate all ground-truth duals; errors in the tool would propagate to every benchmark label.
  • ad hoc to paper For NL4OPT, EasyLP, and ComplexOR, the Llama 3.3 generated primal formulations accepted by objective-value match within five retries are mathematically correct.
    Appendix A.3 describes this construction. The paper itself argues in Section 4 that objective-value checks yield false positives, making this a load-bearing and questionable assumption for about 61% of instances.
  • domain assumption Unparseable or crashing LLM outputs are correctly scored as incorrect for dualization ability.
    Section 5 evaluation pipeline counts crashes and parse failures as incorrect; this conflates code-generation skill with mathematical dualization and is especially salient for Mistral-7B with about 23-25% execution accuracy.
  • domain assumption The CGED canonicalization steps preserve exactly the set of valid P2DC convention differences.
    Section 4.1 defines slack-variable elimination and variable-sign canonicalization; correctness of the benchmark's judgments depends on this equivalence being sound and complete.
  • standard math Standard linear programming duality results are assumed.
    Section 3 invokes weak/strong duality and the SOB rule without proof; this is standard background.

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Cite this review

Pith. "Pith review of DualSchool: How Reliable are LLMs for Optimization Education?." pith.science (2026). https://pith.science/paper/Y2JXIK3I

@misc{pith2026250521775,
  author       = {Pith},
  title        = {Pith review of: DualSchool: How Reliable are LLMs for Optimization Education?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y2JXIK3I}},
  note         = {Machine review of arXiv:2505.21775}
}
read the original abstract

Consider the following task taught in introductory optimization courses which addresses challenges articulated by the community at the intersection of (generative) AI and OR: generate the dual of a linear program. LLMs, being trained at web-scale, have the conversion process and many instances of Primal to Dual Conversion (P2DC) at their disposal. Students may thus reasonably expect that LLMs would perform well on the P2DC task. To assess this expectation, this paper introduces DualSchool, a comprehensive framework for generating and verifying P2DC instances. The verification procedure of DualSchool uses the Canonical Graph Edit Distance, going well beyond existing evaluation methods for optimization models, which exhibit many false positives and negatives when applied to P2DC. Experiments performed by DualSchool reveal interesting findings. Although LLMs can recite the conversion procedure accurately, state-of-the-art open LLMs fail to consistently produce correct duals. This finding holds even for the smallest two-variable instances and for derivative tasks, such as correctness, verification, and error classification. The paper also discusses the implications for educators, students, and the development of large reasoning systems.

Figures

Figures reproduced from arXiv: 2505.21775 by the authors.

Figure 1
Figure 1. The P2DC task of DUALSCHOOL: it illustrates the primal-to-dual conversion, the canonical representation of linear programs and the evaluation using CGED, which is the concatenation of the canonicalization step and the Graph Edit Distance comparison. for reasoning models specialized to structured data, a relatively under-studied but extremely valuable competency. Because of the simplicity of P2DC and the availability… view at source ↗
Figure 2
Figure 2. reports the performance of LLMs on the CORRECTION task. All models struggle to reliably repair the incorrect duals, with accuracies below 60% across all models and error types. Similarly to the GENERATION task, the Phi 4 and Llama 3.3 models outperform the others. These uniformly low accuracies – even on error types that are relatively easy to detect as shown in the next section – reveal that CORRECTION is essential… view at source ↗
Figure 3
Figure 3. Accuracy for the VERIFICATION and CLASSIFICATION tasks by model and error type. 6 Conclusion This paper introduced DUALSCHOOL, the first comprehensive benchmark for probing an LLM’s ability to perform and critique primal-to-dual conversions in linear programming. DUALSCHOOL combines four structured tasks (generation, verification, correction, and error classification) with a graph-based correctness detector that goe… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Accuracy for VERIFICATION and CLASSIFICATION DUALSCHOOL tasks when using structured outputs. – in fact 100% – indicating the LLMs reliably produce executable code. However, the dualization accuracy of the produced routines is very low, with only chatGPT 4o achieving a …
Figure 5
Figure 5. Figure 5: The prompt template used in the Section 5 experiments. [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]

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    This allows to consider equivalent lj ≤ a⊤ j x ≤ uj ⇐ ⇒ −uj ≤ −a⊤ j x ≤ −lj and a⊤ j x = bj ⇐ ⇒ −a⊤ j x = −bj

    Constraint nodes have only one feature bj compared to the li, ui in NGED since in CGED, constraints are reformulated toa⊤ j x ≥ bj rather than lj ≤ a⊤ j x ≤ uj. This allows to consider equivalent lj ≤ a⊤ j x ≤ uj ⇐ ⇒ −uj ≤ −a⊤ j x ≤ −lj and a⊤ j x = bj ⇐ ⇒ −a⊤ j x = −bj. Note ...

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    2D LPs: 36 canonical polytopes, each with three distinct objective vectors, ranging from simple shapes (e.g., unit square, triangle) to more complex ones (e.g., hexagon, irregular pentagon)

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    CO Relaxations: Seven families of combinatorial optimization instances are generated using GECO [35]: maximum independent set, multidimensional knapsack, maximum cut, maximum clique, minimum vertex cover, packing, and production planning

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    know” how to dualize an LP, the authors prompt and manually evaluate each model’s response to “How do you convert a primal linear program to its dual?

    LLM4OPT-Derived LPs: • NLP4LP ([8]): use the provided gurobipy code directly. • NL4OPT[2], Easy LP[4], ComplexOR[3]: these benchmarks only supply an objective value and prompt. Thus, Llama 3.3 is used, following [ 8], to generate gurobipy formulations for each sample. These fo...

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Reviewed August 7, 2026 · model on record in the stance chip above.