GPT-5 Pro Achieves 1.5/L Bound in Convex Optimization, Expert Highlights Speed Over Novelty

A recent development in artificial intelligence has seen GPT-5 Pro generate a new mathematical proof in the field of convex optimization, achieving a bound of η ≤ 1.5/L. This result, confirmed by mathematics researcher Ernest Ryu, surpasses an initial human-discovered bound of η ≤ 1/L from an existing paper. The achievement underscores the evolving capabilities of large language models in complex problem-solving.

Ernest Ryu, whose own research focuses on convex optimization, described the AI's performance as "exciting and impressive." However, he provided a "nuanced take" on the significance of the discovery. While GPT-5 Pro's proof improved upon the original paper's finding, it remains "weaker" than a subsequent human-derived proof which established a tighter bound of η ≤ 1.75/L.

Ryu clarified that the problem, which involved leveraging Nesterov Theorem 2.1.5, primarily required combining known inequalities rather than inventing new foundational mathematical concepts. He noted that an experienced Ph.D. student could typically work out such a proof within a few hours. In stark contrast, GPT-5 Pro accomplished the task with only about 30 seconds of human input, showcasing remarkable computational efficiency.

This incident highlights the potential of advanced AI models like GPT-5 Pro as powerful research assistants, capable of rapidly performing complex calculations and combinatorial searches that would take human experts significantly longer. Despite its impressive speed, Ryu emphasized that the AI is "by no means exceeding the capabilities of human experts" in terms of creative mathematical discovery. The development suggests a future where AI can significantly accelerate the calculational aspects of mathematical research, allowing human researchers to focus more on the creative and conceptual challenges.