PaperlayerView exact revision on arXiv

Source-linked AI summary

Learning to Optimize through Solver-Grounded Self-Play

Xia Jiang, Yaoxin Wu, Chenyu Zhou, Mengzhu Xu, Wim P. M. Nuijten, Yingqian Zhang

arXiv:2609.34205v1cs.LG

TL;DR

Optimization-modeling training often depends on costly, curated examples, raising concerns about generalization and dependence on annotator or teacher capability. OPT-Zero trains one LLM through solver-grounded self-play, alternating between proposing and solving problems. The authors report competitive performance without curated data and generalizability across settings.

  • Problem

    Optimization-modeling training depends on scarce, costly curated examples that may constrain generalization across conventions and solver syntax and anchor reasoning to annotator or teacher proficiency.

    Source: §1 · p. 2

  • Method

    OPT-Zero alternates one LLM between proposing optimization problems and solving them, using external solver verification as reinforcement-learning feedback.

    Source: §1 · p. 3, §3 · p. 4, §3.2 · p. 5

  • Results

    Without curated training data, OPT-Zero matches or surpasses state-of-the-art data-dependent methods and shows strong generalizability across settings.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The authors present self-play as a scalable training paradigm for advancing LLMs in optimization modeling and solving.

    Source: §1 · p. 3

Abstract

from arXiv · show

Optimization modeling is central to many decision-making scenarios, but traditionally requires extensive domain expertise. While Large Language Models (LLMs) have shown promise in automating this process, current training paradigms mainly rely on human-annotated or teacher-generated datasets. This dependence introduces a Generalization Ceiling, where models overfit to narrow data distributions, and Capability Anchoring, where models' reasoning is bounded by annotator proficiency and teacher model capability. In response, we propose OPT-Zero, the first fully self-play training framework for optimization modeling that requires zero external training data. OPT-Zero employs a single LLM in a dual-role closed loop: a Proposer that synthesizes increasingly challenging optimization problems alongside their mathematical formulations and solving code, and a Solver that attempts to resolve the problems given only natural-language problem descriptions. Grounded in execution feedback from external optimization solvers, we alternately train both roles using reinforcement learning. This process fosters an auto-curriculum in which the Proposer and Solver co-evolve: generating harder valid problems by the Proposer seamlessly enhances the structural reasoning ability of the Solver. Extensive results indicate that with zero curated data, OPT-Zero matches state-of-the-art data-dependent methods while exhibiting substantially stronger generalizability, establishing self-play training as a highly scalable paradigm for advancing LLM reasoning in modeling and solving optimization problems.

1 Introduction

OPT-Zero addresses the dependence of optimization-modeling training on costly curated supervision with a solver-grounded self-play loop. The authors report that it matches or surpasses state-of-the-art data-dependent methods without curated training data while generalizing across settings.

1 Introduction

  • LLM optimization modeling has shifted from prompt-based workflows toward fine-tuning on domain-specific datasets.Prompt-based methods operate with frozen parameters; training-based approaches depend on curated examples. [AhmadiTeshnizi et al., 2024; Astorga et al., 2025; Wang et al., 2025]Source: §1 · p. 2
  • Curated supervision requires diverse problem descriptions paired with valid formulations and solver-verified solutions, making high-quality training examples scarce and costly to annotate.Source: §1 · p. 2
  • Static datasets risk a Generalization Ceiling across modeling conventions and solver syntax, and Capability Anchoring to annotator or teacher proficiency.Source: §1 · p. 2
  • OPT-Zero alternates one LLM between proposing challenging problems and solving them, using external solver execution as reinforcement-learning feedback.Source: §1 · p. 3
  • The authors present OPT-Zero as the first fully self-play fine-tuning framework for optimization modeling that requires no external training data.It avoids human annotation and teacher-LLM distillation. [Yang et al., 2025b; Jiang et al., 2025a; Huang et al., 2025]Source: §1 · p. 3
  • Without curated training data, OPT-Zero matches or surpasses state-of-the-art data-dependent methods and shows strong generalizability across settings.Source: §1 · p. 3

2 Related Work

Prior LLM optimization-modeling work includes prompt-based and training-based approaches, while self-play research has explored automatic verification and self-generated curricula. OPT-Zero combines two optimization-specific roles with continuously generated problems and solver-based rewards.

2 Related Work

  • Prompt-based optimization methods use structured workflows, multi-agent collaboration, search, or retrieval, while training-based approaches fine-tune on domain-specific datasets.Prompt-based methods retain frozen parameters, and training-based methods depend on high-quality data. [Wasserkrug et al., 2024; Chen et al., 2024a; Xiao et al., 2024; AhmadiTeshnizi et al., 2024; Deng et al., 2024; Zhang et al., 2025; Astorga et al., 2025; Jiang et al., 2025c; Liu et al., 2025; Huang et al., 2025; Jiang et al., 2025a; Chen et al., 2025a; Ma et al., 2026; Zhou et al., 2026; Xiao et al., 2026; Zhao et al., 2026]Source: §2 · pp. 3–4
  • Prior LLM self-play work uses model interaction or self-generated curricula, including RL with automatic verification in reasoning-intensive domains.[Chen et al., 2024b; Haluptzok et al., 2023; Wilf et al., 2025; Lin et al., 2025; Guo et al., 2025; Huang et al., 2026; Wu et al., 2025; Jayalath et al., 2025; Zhao et al., 2025]Source: §2 · p. 4
  • OPT-Zero separates learning into a Proposer and Solver, continuously generates problems with difficulty-aware sampling, and grounds both roles in solver verification for RL rewards.Source: §2 · p. 4

3 Methodology

OPT-Zero trains one shared LLM as both a Proposer and a Solver in a solver-grounded self-play loop. Its rewards assess problem structure, model and code correctness, and Solver performance; the training loop retains informative problems and refreshes their difficulty as the Solver evolves.

3 Methodology

  • Shared weights let Proposer and Solver improve through a mutually reinforcing loop: harder proposals expose the Solver to richer scenarios, while Solver reasoning supports more valid problems.The Proposer supplies formulations and code; the Solver receives only natural-language descriptions and independently reconstructs both.Source: §3 · p. 4

3.1 Problem Formulation

  • An instance is represented by its optimization sense, objective, variables, constraints, and natural-language scenario; independent solver verification supplies its optimal objective value.The Solver’s declared model is kept separate from the Proposer’s formulation, and generated code is checked against the verified objective.Source: §3.1 · pp. 4–5

3.2 Proposer Reward Specification

  • The Proposer reward gates validity, code correctness, and structural informativeness, scoring structure through variable use, constraint count, coupling, coverage, and connectivity.Its structural target tracks the Solver’s empirical capability frontier rather than requiring difficulty to rise by simply enlarging instances.Source: §3.2 · pp. 5–6

3.3 Solver Reward Specification

  • The Solver receives separate model and code scores: model checks include objective correctness, while code is executed and compared with the verified objective.A 0.1 floor reward for failed code execution avoids discouraging code generation; the two verification paths encourage consistent modeling and coding.Source: §3.3 · p. 6

3.4 Self-Play Training Loop

  • GRPO updates both roles using group-relative advantages, token-level importance ratios, clipping, and a KL penalty against a reference policy.Group-relative rewards address prompt-dependent reward scales; clip-higher permits larger updates on high-reward completions, and identical-reward groups are discarded.Source: §3.4 · pp. 6–7
  • The loop trains the Proposer, pre-scores candidate problems with Solver rollouts, and buffers valid instances that are neither unsolved nor already mastered.Solver training combines 80% newly retained problems with 20% replay; sampling favors harder instances, and difficulty estimates are refreshed after training.Source: §3.4 · pp. 7–8

4 Experiments

Across six benchmarks, OPT-Zero performs competitively with state-of-the-art methods without expert-annotated fine-tuning, and the experiments examine component contributions and generalization across solvers and models.

4 Experiments

  • The study evaluates Pass@1 on six datasets and compares zero-shot LLMs, inference-time methods, and fine-tuned models.OPT-Zero is instantiated from Qwen3-8B-Base; the evaluation reports per-dataset results and macro- and micro-averages.Source: §4 · p. 8

4.1 Main Results

  • Without expert-annotated fine-tuning, OPT-Zero surpasses most data-dependent approaches and performs comparably to state-of-the-art methods, with new fine-tuned-model best results on IndustryOR (54.8) and ReSocratic (89.8).OPT-Zero-D, further fine-tuned with a small amount of problem-description and optimal-objective data, surpasses prior methods on micro-average performance.Source: §4.1 · p. 8, Table 2 · p. 9

4.2 Ablation Studies

  • Removing major components or replacing structured rewards with binary rewards consistently reduces performance; Solver training and its structured reward have the largest accuracy effects.Proposer training and its reward also improve Solver performance, while the reference buffer supports harder Proposer problems and Solver learning; further ablations examine sampling weights and Proposer structural reward.Source: §4.2 · p. 9, Table 3 · p. 9

4.3 Discussion

  • Across four solvers, OPT-Zero improves performance when trained with the target solver, and the study also examines zero-shot transfer to other solvers.Figure 3 compares OPT-Zero with Qwen3-8B-Base; filled markers denote OPT-Zero and hollow markers the base model.Source: §4.3 · p. 9, Figure 3 · p. 10
  • 75.1 vs. 81.6 Macro Avg.: Solver-only training trails joint training, while Solver-only training already improves Proposer correctness and joint training achieves the best Proposer performance.The authors attribute Solver gains from Proposer training to more informative generated problem instances.Source: §4.3 · p. 10, Figure 4 · p. 10
  • OPT-Zero’s gains persist across LLM architectures and sizes, suggesting they are not tied to one base model.The authors describe its supervision as adapting to the model’s current modeling frontier and mitigating anchoring to a static dataset.Source: §4.3 · p. 10

5 Conclusion

OPT-Zero uses a self-play loop between Proposer and Solver roles to train optimization reasoning without human-annotated data. Experiments show competitive benchmark performance, while future work targets broader problem classes and refined rewards.

5 Conclusion

  • OPT-Zero trains Proposer and Solver roles iteratively through reinforcement learning without human-annotated data.The paper reports competitive performance among LLM-based frameworks and significant gains over the backbone on diverse benchmarks.Source: §5 · p. 10
  • The authors identify nonlinear and stochastic programming, refined reward mechanisms, and solver-calling code with automated heuristic discovery as future directions.Source: §5 · p. 10

SUPPLEMENTARY MATERIAL

The supplementary material describes solver-grounded verification, the Proposer and Solver prompts, and a reward combining formulation validity, code correctness, and structural quality. It also explains how the Proposer adapts target complexity based on recent medium-difficulty problems to estimate the Solver’s capability frontier.

A Canonical Verification Pipeline

  • The verification pipeline parses and normalizes model fields, rejects non-linear objectives or constraints, and checks models with Gurobi; generated code runs in a timed subprocess and its objective is compared with the canonical value.Models returned as infeasible or unbounded are invalid; code is valid only if it exits successfully and prints an objective_value result.Source: A Canonical Verification Pipeline · p. 15

B LLM Prompts

  • The first training iteration uses a Seed Proposer Prompt, while later iterations use a Proposer Prompt because the reference-problem buffer is initially empty.Source: B LLM Prompts · p. 15

Seed Proposer Prompt

  • The seed prompt asks the Proposer to create a challenging, solvable linear optimization problem and provide a feasible formulation, matching executable solver code, and a consistent real-world story.It restricts models to IP, LP, or MILP and asks for a concrete feasibility check.Source: Seed Proposer Prompt · pp. 15–16

Proposer Prompt

  • The regular Proposer prompt requires a feasible, bounded, linear model with matching solver code and story, and asks for greater model complexity and diversity from buffered references.It also calls for a surprising story domain distinct from the reference stories.Source: Proposer Prompt · p. 16

Solver Prompt

  • The Solver prompt asks the model to derive an optimization formulation and solver script from a problem description, explicitly checking variables, bounds, objective, constraints, feasibility, and boundedness before implementation.Source: Solver Prompt · pp. 17–18

C Proposer Reward Details

  • The Proposer reward multiplies validity, code correctness, and structural quality; structural scoring uses deterministic formulation features and adaptive targets rather than online Solver solve rates.Its five structural sub-scores assess active-variable use, unique non-trivial constraints, coupling density, variable coverage, and constraint-graph connectivity; the approach avoids the cost of multiple Solver rollouts per Proposer completion.Source: C Proposer Reward Details · pp. 19–20
  • Targets update from problems with solve rates in (0.25, 0.75]; if fewer than three qualify, they remain unchanged.Targets start at (5, 5, 2.0); with at least three qualifying problems, variable and constraint targets use their respective medians, density is their ratio, and all three are smoothed with αEMA = 0.3 and bounded to v∈[4, 80], c∈[3, 30], and d∈[2.0, 10.0].Source: C Proposer Reward Details · p. 20

D OPT-ZERO Self-Play Training Algorithm

OPT-Zero alternates Proposer and Solver training in a closed loop, using generated problems and solver feedback to adapt the training set.

D OPT-ZERO Self-Play Training Algorithm

  • Each iteration trains the Proposer with reference problems from the current buffer, then generates and canonically verifies candidate problems.Invalid, infeasible, or unbounded problems are discarded; valid candidates are pre-scored using Solver rollouts.Source: D OPT-ZERO Self-Play Training Algorithm · p. 20
  • The algorithm retains problems with empirical solve rates strictly between 0 and 1, then deduplicates and evicts buffer entries by signature and capacity.The initial buffer is empty, and the buffer also supports replay in later iterations.Source: D OPT-ZERO Self-Play Training Algorithm · p. 20
  • Solver training mixes newly generated problems with replayed buffer problems at an 80%/20% split and uses difficulty-aware sampling.Replayed problems have their solve-rate estimates and difficulty labels refreshed.Source: D OPT-ZERO Self-Play Training Algorithm · p. 20

E Detailed Experimental Settings

The settings specify the self-play loop, training configuration, benchmark sources, and objective-based evaluation and aggregation procedures.

E Detailed Experimental Settings

  • Training uses one server with 4 NVIDIA H100 GPUs and 32 CPU cores, bfloat16, and DAPO-style GRPO with a 1 × 10^-6 learning rate and KL coefficient 0.05.The setup also uses full-parameter training, dynamic sampling, token-level loss normalization, and eight completions per prompt group.Source: E Detailed Experimental Settings · pp. 20–21

D OPT-ZERO Self-Play Training Algorithm

  • The loop seeds an empty buffer, trains the Proposer with sampled reference problems, verifies candidates, and pre-scores valid problems with Solver rollouts.Informative candidates with solve rates between 0 and 1 are retained, with signature-based deduplication and capacity-based eviction.Source: E Detailed Experimental Settings · p. 21
  • Solver training draws 80% of samples from newly generated problems and 20% from replay, using difficulty weights of 0.2, 0.3, and 0.5 for easy, medium, and hard problems.Replayed problems have their solve-rate estimates and difficulty labels refreshed.Source: E Detailed Experimental Settings · p. 21

E Detailed Experimental Settings

  • The six-benchmark suite uses cleaned public test subsets totaling 1,492 problems, including 403 ReSocratic instances.The cited table is identified as listing the cleaned test subsets used in the evaluation.Source: E Detailed Experimental Settings · p. 21, Table 4 · p. 22
  • Single-generation evaluation scores objective-value-based Pass@1, requiring successful execution within 60 seconds and objective matching at 5% relative tolerance, with 10^-5 absolute tolerance near zero.Training-time verification instead uses a 1% relative tolerance.Source: E Detailed Experimental Settings · p. 22
  • Macro-average weights benchmarks equally, micro-average weights problems equally, and baseline results from original publications are marked with an asterisk in Table 2.On the 42-instance IndustryOR set, one additional correct prediction changes accuracy by approximately 2.38 percentage points.Source: E Detailed Experimental Settings · p. 22

E.1 Training Cost and Inference Throughput

Across the reported evaluations, OPT-Zero improves general-purpose reasoning and coding scores, transfers from Gurobi training to other solver interfaces, and benefits from inference-time scaling and hard-instance sampling. The examples show increasingly structured generated problems, while the training and throughput measurements quantify the computational cost.

E.1 Training Cost and Inference Throughput

  • OPT-Zero's Qwen3-8B self-play run uses approximately 292 allocated H100 GPU-hours, about 24% less than SIRL-7B's reported 384 GPU-hours.OPT-Zero's allocation includes self-generated curriculum construction; SIRL's figure covers its two training stages.Source: §E.1 · pp. 22–23
  • Generating and verifying 16 candidates for each of 1,492 problems takes 3,985 seconds total, or approximately 2.67 seconds per problem.Generation takes 3,813 seconds and parallel verification takes 172 seconds on one H100.Source: §E.1 · p. 23, Table 6 · p. 23

F.1 Example Proposer Generations

  • Across iterations, Proposer examples progress from simpler tasks to richer stories with more variables and coupled constraints, while remaining valid and semantically coherent.The authors interpret this as an auto-curriculum that expands problem complexity and supports Solver improvement without external annotated data.Source: §F.1 · p. 23, Table 7 · p. 24

F.2 Example Solutions

  • On a MAMO ComplexLP example, OPT-Zero-8B formulates integer food counts and returns the optimal objective value 32, whereas Qwen3-8B-Base uses binary decisions and fails to produce a feasible solution.The authors attribute the trained model's result to learning variable types and solving through trial-and-error reinforcement learning.Source: §F.2 · p. 25

F.3 General-purpose Problem-solving Capability

  • OPT-Zero-8B scores highest among three models on all four general reasoning and coding tasks, including 80.5% HumanEval Pass@1 and 88.3% GSM8K accuracy.The comparisons are 80.5% versus 18.9% for StepORLM-8B on HumanEval and 88.3% versus 20.9% for LLMOPT-14B on GSM8K.Source: §F.3 · p. 28, Table 8 · p. 28

F.4 Zero-shot Solver Transferability

  • Gurobi-trained OPT-Zero-8B outperforms listed baselines in every reported comparison across CPLEX, SCIP, and Pyomo.The evaluation on those interfaces is out of distribution for the Gurobi-trained model.Source: §F.4 · p. 28, Table 9 · p. 29

F.5 Inference-time Scaling

  • OPT-Zero has higher Pass@N coverage at small candidate counts and retains an advantage on harder benchmarks as N increases; one reflection round also consistently improves performance.On NLP4LP, OPT-Zero reaches 100% Pass@4; reflection conditions a revision on execution feedback without exposing the correct formulation or objective.Source: Figure 6 · p. 29, Figure 5 · p. 29, §F.5 · p. 30

F.6 Effect of Difficulty-Aware Sampling

  • Sampling more hard instances yields the best performance, while prioritizing easier instances causes larger declines on difficult benchmarks such as ComplexLP and IndOR.The reported easy-, medium-, and hard-first weights are (0.5, 0.3, 0.2), (0.3, 0.5, 0.2), and (0.2, 0.3, 0.5), respectively.Source: §F.6 · p. 30, Figure 7 · p. 30

F.7 Comparison to Training with Solve-rate Reward

The comparison favors structural-reward GRPO over an offline solve-rate-based DPO baseline across six benchmarks. OPT-Zero also improves results across four model backbones, including a strong instruction-tuned initialization.

F.7 Comparison to Training with Solve-rate Reward

  • Structural-reward GRPO outperforms solve-rate-based DPO on all six benchmarks, with ComplexLP accuracy of 58.6% versus 29.7%.The authors' method uses optimization-specific structural feedback and obtains Proposer rewards from formulation and code verification.Source: §F.7 · p. 31, Table 10 · p. 31

F.8 Fine-tuning Different LLMs

  • OPT-Zero improves all four tested backbones; Gemma4-31B-IT gains 15.6 Macro and 12.4 Micro percentage points from its instruction-tuned initialization.The reported before/after Macro/Micro scores are 73.5/80.4 and 89.1/92.8.Source: §F.8 · p. 31, Table 11 · p. 31
  • Across model families, results support broad applicability across architectures, sizes, and initializations; Gemma4-31B-IT improves from 73.5/80.4 to 89.1/92.8 Macro/Micro, gaining 15.6 Macro and 12.4 Micro percentage points from a strong instruction-tuned initialization.The framework supports both full-parameter training and parameter-efficient adaptation across different model families.Source: §F.8 · p. 31

F.9 Transfer to Challenging External Benchmarks

OPT-Zero improves Gemma4-31B-IT accuracy on two additional held-out benchmarks, including industrial-scale MILP tasks. The curriculum analysis describes how verified Proposer candidates are filtered and sampled, and connects curriculum components to held-out performance.

F.9 Transfer to Challenging External Benchmarks

  • OPT-Zero raises Gemma4-31B-IT accuracy from 41.0% to 48.2% on OptMATH-Bench and from 8.92% to 11.74% on MIPLIB-NL.The held-out tests include natural-language modeling of industrial-scale mixed-integer linear programs.Source: §F.9 · p. 32, Table 12 · p. 32

F.10 Empirical Difficulty of the Self-generated Curriculum

  • The Solver fails on all eight attempts for 55.7% of verified candidates, while 32.3% yield both successful and unsuccessful attempts.Candidates are evaluated with eight contemporaneous Solver rollouts; the Proposer generates a formulation, implementation, and story, while the Solver reconstructs from the story.Source: §F.10 · p. 32, Table 13 · p. 32
  • The solve-rate filter retains candidates with 0 < ρ < 1, then training samples easy, medium, and hard problems with weights (0.2, 0.3, 0.5).Candidates with zero or eight successful rollouts are excluded at filtering; structural deduplication and capacity control also govern buffer admission.Source: §F.10 · pp. 32–33
  • Removing Proposer training lowers Macro/Micro accuracy from 81.6/90.3 to 75.1/84.3, while removing the reference buffer lowers it to 72.9/83.2.The authors connect these component ablations to the role of Proposer training and buffer-based task reuse in learning and external task performance.Source: §F.10 · p. 33

F.11 Combining Self-play Training with Chain-of-Experts Inference

CoE inference improves both the base and trained models, while single-generation OPT-Zero already exceeds the CoE-enhanced base model. An ablation also finds lower Micro accuracy when Proposer code generation and its correctness reward are removed.

F.11 Combining Self-play Training with Chain-of-Experts Inference

  • With CoE, OPT-Zero-8B reaches 82.8 Macro and 91.8 Micro accuracy, leading the CoE-enhanced Qwen3-8B-Base by 11.6 and 11.4 percentage points.The same CoE procedure uses approximately eight LLM calls; single-generation OPT-Zero-8B reaches 81.6 Macro and 90.3 Micro with one call per problem.Source: §F.11 · p. 33, Table 14 · p. 33

F.12 Contribution of Proposer Code Generation

  • Removing Proposer code generation and its code-correctness reward reduces Micro-average accuracy from 90.3% to 87.1%, a 3.2-point decrease.The Solver retains its formulation and executable-code outputs and original reward in this ablation.Source: §F.12 · p. 34, Table 15 · p. 34
  • The Proposer's code branch adds formulation-to-implementation practice and code-correctness feedback alongside the Solver's description-to-formulation-and-code task.Both roles update the same model, and canonical validity and code consistency provide feedback for these complementary tasks.Source: §F.12 · pp. 33–34

F.13 Textual Diversity of Generated Training Problems

OPT-Zero’s sampled self-play problems show greater textual diversity and less repetitive wording than the sampled OptMATH-Train problems. The section also situates OPT-Zero’s adaptive, solver-verifiable curriculum among corpus-based and self-play approaches and reports results across backbones and interfaces.

F.13 Textual Diversity of Generated Training Problems

  • The OPT-Zero buffer sample scores 244 versus 178 on Vendi Score and 0.32 versus 0.56 on nearest-neighbor redundancy, indicating greater textual diversity and less repetitive wording than OptMATH-Train.Each sample contains 300 problems; Vendi Score summarizes diversity across the similarity spectrum, while nearest-neighbor redundancy measures similarity to the closest other statement.Source: §F.13 · p. 34, Table 16 · p. 34

F.14 Adaptive Supervision and Relation to Prior Work

  • The self-play curriculum continually generates and selects tasks as the learner’s capabilities change, complementing reusable synthetic corpora.The problems are generated by the model and receive feedback from optimization solvers.Source: §F.14 · p. 35
  • OPT-Zero brings task-generating self-play to end-to-end optimization modeling by combining canonical formulation solving, executable-code feedback, optimization-specific structural rewards, and a difficulty-aware reference buffer.The comparison describes OptMATH as corpus-based, SIRL as applying solver feedback to external problems, and StepORLM as co-evolving a policy and process verifier using a synthesized set.Source: §F.14 · p. 35, Table 17 · p. 35
  • OPT-Zero-D draws 30% of Solver training problems per iteration from Chen et al. (2025a), and its Table 2 results demonstrate compatibility between self-generated supervision and external training data.Source: §F.14 · p. 35
  • Experiments evaluate training with different solver interfaces and Gurobi-trained weights on new interfaces to assess adaptability and zero-shot transfer.The interfaces named are DOcplex, PySCIPOpt, and Pyomo.Source: §F.14 · p. 35
  • Qwen3.5-4B rises from 16.8% to 89.9% Micro-average, while Gemma4-31B-IT rises from 80.4% to 92.8% using rank-128 LoRA.The passage reports improvements across four backbones, using full-parameter or parameter-efficient training.Source: §F.14 · p. 35

G Frequently Asked Questions

The FAQ clarifies that OPT-Zero’s main configuration uses self-play rather than external optimization training data, while retaining a pretrained model as its starting policy. It explains the solver-verification environment, LP/IP/MILP scope, story-to-formulation checks, and the reported contribution of training components.

G Frequently Asked Questions

  • The main configuration uses no external optimization corpus, human-annotated formulations, expert-written solution programs, or teacher-generated solutions; a pretrained model supplies the starting policy.Solver-grounded interaction produces subsequent training supervision.Source: G Frequently Asked Questions · p. 36
  • LP, IP, and MILP support automatic parsing, solving, and evaluation because mature exact solvers provide feasibility, boundedness, and objective-value checks.These classes are also common in optimization-modeling benchmarks and prior training-based methods.Source: G Frequently Asked Questions · p. 36
  • The stated extension boundary is nonlinear, stochastic, robust, and multi-objective optimization, which requires suitable problem representations and reliable solver feedback.Source: G Frequently Asked Questions · p. 36
  • Story–formulation alignment is checked through canonical solving and code verification, while the prompt and Solver reconstruction task encourage the story to preserve the model’s details.The Proposer’s story is required to preserve variables, coefficients, bounds, objective, and constraints; the Solver receives only the story and reconstructs a formulation and executable code.Source: G Frequently Asked Questions · p. 36
  • Invalid canonical formulations receive zero validity reward; valid formulations with incorrect Proposer code receive a reduced correctness factor, and solve-rate filtering plus structural deduplication select buffer problems.The Solver’s empirical solve rate guides curriculum selection, and the buffer retains candidates with mixed outcomes.Source: G Frequently Asked Questions · p. 36
  • Removing Solver training, Proposer training, or the reference buffer, or replacing structural rewards with binary rewards, reduces performance in the reported ablations.The paper attributes effective self-play support to the verifier, structural rewards, and reference buffer.Source: G Frequently Asked Questions · p. 36
Loading 2609.34205v1…