Max It or Miss It: Benchmarking LLM On Solving Extremal Problems

AI summary

Overview

• Research area: Evaluation and benchmarking of large language model reasoning, specifically mathematical optimization reasoning (finding extrema under constraints).
• Technical level: Intermediate — the benchmark construction is conceptually simple to follow, but interpreting the results assumes familiarity with LLM benchmarking, chain-of-thought reasoning, and olympiad-style inequalities.
• Scope: The paper introduces ExtremBench, a 93-problem benchmark of extremal (max/min) problems converted from Chinese Mathematical Olympiad inequality exercises, and uses it to evaluate a range of open-source reasoning models against AIME25 performance.

What This Paper Is About

Existing math benchmarks such as GSM8K, MATH-500, and AIME focus mainly on algebraic manipulation and arithmetic, and barely test optimization reasoning — the ability to find extrema under constraints. The authors argue that this skill is fundamental to planning, control, resource allocation, and prompt search, yet it is not systematically measured. Their goal is to build a benchmark that isolates extremal-solving ability and to check whether strong general math performance actually predicts success on these problems.

Key Contributions

1. ExtremBench construction: The authors curated problems from An Introduction to the Proving of Elementary Inequalities, a collection of Chinese Mathematical Olympiad (CMO) exercises, translated them from Chinese to English, and converted proof-style inequalities ("prove that A ≤ B") into equivalent extrema-finding tasks ("find the minimum/maximum of A − B"). From 100 initial problems, 93 were retained after filtering out strict inequalities where equality cannot be achieved.
2. A verifiable format for proof-based math: The transformation preserves the reasoning difficulty of the original inequality proof but produces a single numerical answer, enabling automatic evaluation. The authors frame this as a way to turn hard-to-verify proofs into numerically verifiable problems usable for training and reward modeling.
3. Broad model evaluation: They evaluate Qwen3 (1.7B, 4B-Thinking-2507, 8B, 14B, 32B, 30B-A3B-Thinking-2507, 235B-A22B-Thinking-2507), GPT-OSS (20B, 120B), and DeepSeek-R1 (1.5B, 7B, 8B, 14B), reporting average performance over 3 repeated trials.
4. Evidence of a benchmark blind spot: The results show extremal-solving ability does not consistently track performance on AIME25 or MATH-500, motivating domain-specific benchmarks.

Main Findings

• ExtremBench composition: The final dataset contains 93 extremal problems — 62 minimization problems and 31 maximization problems — each with constraint specifications, an objective function derived from the original inequality, and a numerical answer.
• Strong AIME25 performance does not transfer: GPT-OSS-120B-High and GPT-OSS-20B-High score above 90% on AIME25 but plateau around 70% on ExtremBench, indicating that strong general mathematical reasoning does not guarantee proficiency in optimization tasks.
• Scale is not a reliable predictor: Qwen3-14B matches the ExtremBench performance of Qwen3-235B despite having 17 times fewer parameters, suggesting extremal-solving ability may depend more on training data or architectural choices than on raw model size.
• Thinking variants lead on ExtremBench: Qwen3-Thinking variants achieve the strongest ExtremBench performance at 75–80%, despite only moderate AIME25 scores.
• DeepSeek-R1 models trail on both: DeepSeek-R1 models show consistently lower performance, scoring 50–60% on ExtremBench and 30–40% on AIME25.
• General conclusion: Extremal problem-solving appears to be a distinct mathematical competency that current benchmarks fail to capture, with some models strong at general math but weak at extremal reasoning, and vice versa.

Methodology in Plain English

The pipeline has four steps. First, the authors took a set of CMO inequality exercises written in Chinese and translated them into English with an LLM, with bilingual authors verifying mathematical accuracy. Second, they used Anthropic's Claude Opus 4.1 with a fixed prompt to rewrite each proof problem as an extremum problem: keep all original conditions, move constant terms to the other side, and ask for the extremum of A − B, output only as LaTeX. Third, each converted problem was manually verified by the authors to confirm (1) all original constraints were preserved, (2) the extremal formulation is mathematically equivalent to the original inequality, and (3) the statement is unambiguous and self-contained. Problems were then filtered from 100 down to 93 by removing strict inequalities where equality cannot be achieved. Fourth, models were run using SGLang on NVIDIA B200 GPUs, with answers checked by the Math-Verify library, and each configuration was run 3 times and averaged. Results were compared against published AIME25 scores for the same models.

Why This Matters

For research, the paper challenges the assumption that a high score on general math benchmarks implies broad mathematical competence. It shows that benchmark choice can materially change which models appear strongest, and it offers a reusable recipe for converting proof-style problems into automatically gradable ones — relevant to anyone building training data or reward signals for reasoning models.

Real-world applications of the underlying capability, as the authors frame them:

• Planning and control, where an agent must select actions that are optimal under constraints.
• Resource allocation, where budgets or capacities must be distributed to maximize or minimize an objective.
• Hyperparameter tuning in machine learning, a constrained search for an optimum.
• Economic equilibrium problems and prompt search, both described as optimization-under-constraints tasks.

Industry relevance: teams that select models based on leaderboard math scores may be picking the wrong model for optimization-heavy workloads. The paper also matters for reinforcement learning pipelines that depend on verifiable answers, since it demonstrates a way to expand the supply of verifiable training problems from proof-based sources.

Future Directions

• Extend the transformation paradigm: The authors propose converting existence proofs in combinatorics into counting problems, congruence proofs in geometry into distance optimization, and convergence proofs in analysis into rate-of-convergence optimization, potentially unlocking large repositories of proof problems for RL training and reward modeling.
• Broaden the benchmark's mathematical coverage: Add multi-objective optimization, constrained optimization with equality constraints, and discrete optimization problems to test optimization reasoning across different mathematical structures.
• Study the mechanism behind the discrepancy: Investigate why general mathematical reasoning and extremal-solving ability diverge, which could reveal how LLMs encode and apply different types of mathematical knowledge and inform targeted training.
• Target extremal reasoning directly: Develop supervised fine-tuning or RL approaches aimed specifically at extremal reasoning, possibly using curriculum learning that progressively increases constraint complexity and dimensionality.
• Test predictive value downstream: Examine whether extremal-solving capability is a better predictor of performance in scientific computing, operations research, and automated theorem proving, where optimization reasoning is fundamental.

Target Audience

Researchers and engineers working on LLM evaluation, mathematical reasoning, and reinforcement learning from verifiable rewards will benefit most, particularly those who design benchmarks or build training data from math problems. It is also relevant to practitioners who choose models for optimization-oriented tasks such as planning, scheduling, and resource allocation, and to students of olympiad mathematics interested in how proof problems can be reformulated into computationally checkable ones. Readers without background in LLM benchmarking or inequality problems will find the setup easy to grasp but the results harder to place in context.

Authors’ abstract

Test-time scaling has enabled Large Language Models (LLMs) with remarkable reasoning capabilities, particularly in mathematical domains, through intermediate chain-of-thought (CoT) reasoning before generating final answers. However, the specific sources and mechanisms underlying these reasoning capabilities remain insufficiently understood. Optimization reasoning, i.e. finding extrema under constraints, represents a fundamental abstraction that underpins critical applications in planning, control, resource allocation, and prompt search. To systematically evaluate this capability, we introduce ExtremBench, a benchmark dataset for solving mathematical extremal problems, curated from inequality exercises used for Chinese Mathematical Olympiad and transformed into $93$ standardized extrema-finding problems. We conduct extensive evaluations across various state-of-the-art open-source model families, including the Qwen3, GPT-OSS, and DeepSeek. Our results reveal that LLMs' extremal-solving reasoning capabilities do not always align with those of current mathematical benchmarks such as AIME25 and MATH-500, with some models showing strong general mathematical reasoning but poor extremal-solving skills, and vice versa. This discrepancy highlights a critical gap in current evaluation practices and suggests that existing benchmarks may not comprehensively capture the full spectrum of mathematical reasoning abilities.