Point Convergence of Nesterov's Accelerated Gradient Method
Nesterov's accelerated gradient method (1983) is a cornerstone of optimization, yet whether its iterates themselves converge to a minimizer, rather than just the function values, stayed open for over forty years. Jang and Ryu resolve it in the affirmative. Ryu first announced the continuous-time result on X; Bot, Fadili and Nguyen's concurrent human proof of the critical-regime case (answering a decade-old conjecture of Attouch and co-authors) explicitly credits that AI-assisted announcement as what it discretizes.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Argument
- Field
- Convex optimization
- Posed by
- Yurii Nesterov (method); point convergence open since its introduction
- Year posed
- 1983
- Years open
- 42y
- Solved
- 2025-10-27
- Model
- GPT-5 Pro
- Vendor
- OpenAI
- Collaborators
- Uijeong Jang, Ernest K. Ryu
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 32 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The discovery was heavily assisted by ChatGPT (GPT-5 Pro), and the paper documents how: the process was highly interactive, with roughly 80% of generated arguments incorrect but several ideas novel enough to pursue; the working prompt supplied the continuous-time proof in LaTeX and asked for a discrete-time analogue. The authors note that after the result was found, GPT-5 Pro could reproduce a correct proof from a single well-formulated prompt.
Verification
A v2 preprint submitted for journal review; the result triggered immediate follow-up work (Bot-Fadili-Nguyen, and inexact-FISTA extensions) but no formal review has appeared.