Max It or Miss It: Benchmarking LLM On Solving Extremal Problems
ExtremBench benchmark reveals LLM discrepancies in solving extremal problems.
Key Findings
Methodology
The paper introduces ExtremBench, a benchmark for evaluating LLM optimization reasoning by transforming inequality exercises from the Chinese Mathematical Olympiad into standardized extremal problems. Models like Qwen3, GPT-OSS, and DeepSeek were evaluated.
Key Results
- Result 1: GPT-OSS-120B excels in AIME25 but only achieves 70% accuracy in ExtremBench, highlighting differences between general mathematical reasoning and optimization tasks.
- Result 2: Qwen3-14B matches Qwen3-235B's performance on ExtremBench despite having 17× fewer parameters, indicating training data or architecture choices matter more than scale.
- Result 3: DeepSeek-R1 performs poorly on both benchmarks, showing the uniqueness of extremal problem-solving ability.
Significance
The study reveals that current mathematical benchmarks fail to capture the full spectrum of LLM mathematical reasoning abilities, emphasizing the need for specialized evaluation frameworks to identify distinct reasoning competencies. This is significant for academia and industry, particularly in planning, control, and resource allocation.
Technical Contribution
By converting hard-to-verify inequality proofs into numerically verifiable extremal problems, the paper provides a new method for systematically evaluating optimization reasoning. This transformation offers a new perspective for RL training and could extend to other mathematical domains.
Novelty
This is the first systematic evaluation of LLMs' capabilities in solving extremal problems, revealing significant differences from existing mathematical benchmarks. Compared to related work, the paper provides a new evaluation framework.
Limitations
- Limitation 1: Some models perform well in general mathematical reasoning but poorly in optimization tasks, possibly due to training dataset choices.
- Limitation 2: Current benchmarks fail to fully assess LLM optimization reasoning capabilities.
Future Work
Future research could expand ExtremBench to include multi-objective optimization, constrained optimization, and discrete optimization problems, and explore LLM applications in scientific computing and automated theorem proving.
AI Executive Summary
Current mathematical benchmarks fail to fully assess LLM mathematical reasoning abilities, especially in optimization reasoning. The paper introduces ExtremBench, a benchmark for evaluating LLM optimization reasoning by transforming inequality exercises from the Chinese Mathematical Olympiad into standardized extremal problems. Models like Qwen3, GPT-OSS, and DeepSeek were evaluated, revealing significant differences between general mathematical reasoning and optimization tasks. The study highlights a blind spot in current evaluation practices, emphasizing the need for specialized frameworks to identify distinct reasoning competencies. Future research could expand ExtremBench to include multi-objective optimization, constrained optimization, and discrete optimization problems, and explore LLM applications in scientific computing and automated theorem proving.
Deep Analysis
Background
In recent years, LLMs have made significant advances in mathematical reasoning, particularly in chain-of-thought reasoning. However, existing benchmarks like GSM8K and MATH-500 primarily focus on algebraic manipulation and arithmetic computation, leaving optimization reasoning largely unexplored. Optimization reasoning is crucial for applications in planning, control, and resource allocation, necessitating specialized evaluation frameworks to identify LLM optimization reasoning capabilities.
Core Problem
The core problem is that existing mathematical benchmarks fail to fully assess LLM optimization reasoning capabilities. Extremal problems require the ability to identify constraint boundaries, understand trade-offs between competing objectives, and recognize when optimal solutions occur at critical points or boundaries. These skills are essential for many practical applications.
Innovation
The paper's core innovation is the introduction of the ExtremBench benchmark, which evaluates LLM optimization reasoning by transforming inequality proof problems into extremal problems. This transformation preserves mathematical complexity and reasoning requirements while creating a standardized and easy-to-verify format.
Methodology
- �� Source material from Chinese Mathematical Olympiad inequality exercises.
- �� Transform inequality proof problems into extremal problems while preserving original constraints.
- �� Evaluate multiple LLM models, including Qwen3, GPT-OSS, and DeepSeek.
- �� Manually verify converted problems to ensure mathematical equivalence.
Experiments
The experimental design includes evaluating various LLM models' optimization reasoning capabilities using the ExtremBench benchmark. Models like Qwen3, GPT-OSS, and DeepSeek were used, with inference conducted on NVIDIA B200 GPUs. Average performance from three repeated trials was reported.
Results
Experimental results show GPT-OSS-120B excels in AIME25 but only achieves 70% accuracy in ExtremBench. Qwen3-14B matches Qwen3-235B's performance on ExtremBench despite having 17× fewer parameters. DeepSeek-R1 performs poorly on both benchmarks, showing the uniqueness of extremal problem-solving ability.
Applications
ExtremBench can be used to evaluate LLM optimization reasoning capabilities in planning, control, and resource allocation. This is particularly important for applications requiring constraint boundary identification and understanding of trade-offs.
Limitations & Outlook
Current benchmarks fail to fully assess LLM optimization reasoning capabilities, with some models performing well in general mathematical reasoning but poorly in optimization tasks. Future research could expand ExtremBench to include multi-objective optimization, constrained optimization, and discrete optimization problems.
Plain Language Accessible to non-experts
Imagine you're cooking in a kitchen. You have many ingredients, but to make the perfect dish, you need to find the best combination. Solving mathematical extremal problems is similar; you need to find the optimal solution under given conditions. This research helps computers do just that.
ELI14 Explained like you're 14
Hey, imagine you're playing a game and need to find the best strategy in limited time. Researchers are teaching computers how to find the best answers in complex math problems, just like you find the best way to win a game. This research makes computers smarter, able to solve tougher problems!
Glossary
ExtremBench
A benchmark for evaluating LLM optimization reasoning capabilities, transformed from Chinese Mathematical Olympiad inequality exercises.
Used to assess LLM performance in solving extremal problems.
LLM
Large Language Model capable of complex language understanding and generation.
Used for mathematical reasoning and optimization problem-solving.
Optimization Reasoning
The process of finding extrema under given constraints.
Core capability being evaluated in the study.
Qwen3
A model family used to evaluate LLM optimization capabilities.
Performed well in ExtremBench benchmark.
GPT-OSS
A large language model excelling in general mathematical reasoning.
Performed well in AIME25 benchmark.
Open Questions Unanswered questions from this research
- 1 Current benchmarks fail to fully assess LLM optimization reasoning capabilities, requiring specialized evaluation frameworks.
- 2 How to expand ExtremBench to include multi-objective optimization and discrete optimization problems.
Applications
Immediate Applications
Planning and Control
Using LLM optimization reasoning capabilities to improve planning and control in automated systems.
Long-term Vision
Resource Allocation
Applying LLM optimization reasoning capabilities in complex resource allocation problems to enhance efficiency.
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.