GPT-5.6 Sol Pro solves open problem in convex optimization (medium.com)

OpenAI's new GPT-5.6 Sol model has made a significant breakthrough in mathematical optimization, solving a longstanding problem that has perplexed researchers since 1996. In just a 2.5-hour session, the model produced a proof that establishes a new lower bound for the oracle complexity of deterministic, possibly nonsmooth, zeroth-order convex optimization—a key question relevant to various fields, including machine learning and simulation-based engineering. The result demonstrates that the existing algorithm requiring approximately \(d^2\) function evaluations is, in fact, optimal for these types of problems, effectively closing a gap that had remained open for decades. This development underscores the potential of AI models in advancing mathematical research while also highlighting the need for careful verification of AI-generated results. The author employed the Lean programming language to formally verify the proof, confirming its correctness and showcasing the crucial role formal verification tools will play in the future of mathematical research. As AI becomes an increasingly powerful tool in generating plausible mathematical arguments, integrating verification processes, like those offered by Lean, may become essential to distinguish genuine findings from misleading outputs, marking a notable shift in the landscape of mathematical optimization and research practices.