Mathematician Uses OpenAI's GPT-5 to Solve 40-Year-Old Optimization Problem
Executive Summary
OpenAI has published a case study detailing how UCLA Professor Ernest Ryu used its new GPT-5 model to help solve a decades-old open problem in optimization theory. The model acted as an AI research collaborator, rapidly generating novel ideas and exploring solution paths for the Nesterov Accelerated Gradient (NAG) stability problem. By pairing GPT-5's ability to surface unconventional approaches with his own subject-matter expertise for verification, Ryu was able to achieve a breakthrough in a fraction of the time a traditional research process would have taken.
Key Takeaways
* AI as Research Collaborator: GPT-5 was not used to autonomously solve the problem but as an interactive partner to brainstorm, propose ideas, and explore potential proofs.
* Accelerated Idea Generation: The model significantly sped up the research workflow by quickly proposing and helping to discard numerous approaches, condensing what could have been weeks of work into approximately 12 hours over three days.
* Cross-Disciplinary Insight: A key capability was GPT-5's ability to pull existing tools, equations, and ideas from a massive corpus of papers, including from adjacent mathematical fields that a human expert might not have immediately considered.
* Human Expertise is Critical: Ryu's success depended entirely on his expert intuition to guide the model, identify promising but flawed suggestions, verify all calculations, and develop the final proof. The process required constant human oversight and validation.
* The Breakthrough Moment: The solution's foundation came from a GPT-5 suggestion to restructure the problem's governing equations. While the model's output was not perfectly correct, it provided the crucial structural insight that Ryu then developed into the final, rigorous proof.
Strategic Importance
This announcement positions GPT-5 as a powerful tool for specialized scientific research, moving beyond general-purpose tasks. It showcases a new paradigm of human-AI collaboration that can accelerate innovation and solve complex, long-standing problems in academic and enterprise R&D environments.