Actually, the real open problem is to establish point convergence of the Nesterov accelerated gradient (NAG) method. That is, the discrete-time, implementable algorithm.
2/N
NAG was introduced by Nesterov in 1983 as an accelerated improvement upon plain gradient descent, yet its point convergence remained unresolved until today. Nesterov ODE was studied as a simplified proxy, aimed at gaining insight into the behavior of its discrete counterpart.
3/N
Indeed, translating the continuous-time proof into the discrete-time setup turned out to be relatively straightforward. With one piece of feedback, ChatGPT was able to work out the proof.
chatgpt.com/share/68fc3977-b…
4/N
In 2016, Kim and Fessler introduced a variant called OGM, which improves upon NAG by a constant factor of 2. We also show that OGM exhibits point convergence.
6/N
After the initial tweet went out, several colleagues in the optimization community reached out with excitement and amusement. It was a lot of fun reconnecting with old friends. And I now have some interesting conversational material for the next optimization conference.
7/N
In particular, Radu Ioan Boţ, Jalal Fadili, Dang-Khoa Nguyen reached out with a preprint of their own, capitalizing on the ideas in the continuous-time proof to also establish point convergence of Nesterov 1983 and more!
@RaduIoanBot
8/N
The arXiv preprint of [Bot ̧ Fadili, Nguyen 2025] will go public in a few days. They also prove weak convergence in the infinite-dimensional Hilbert space setting. (ChatGPT will now be happy.)
x.com/ErnestRyu/status/19812…
9/N
They also argue that the FISTA method [Beck, Teboulle 2009] exhibits point convergence.
Check out their work when it goes public! (As far as I know, [Bot ̧ Fadili, Nguyen 2025] did not use AI tools.)
10/N
Oct 25, 2025 · 6:58 AM UTC
Conclusion:
Starting from the key idea of the first tweet, we extended the convergence result to several related settings and resolved the main 42-year-old open problem, with ChatGPT doing most of the heavy lifting along the way.
11/N
Overall, this entire journey took just a week, less than 30 hours of my time. ChatGPT’s assistance provided a significant speedup, and without it, I would most likely have given up after three days of slow progress. (As I did in the past.)
12/N
Again, ChatGPT is now at the level of solving some math research questions, but you do need an expert guiding it.
I strongly encourage fellow mathematicians to try incorporating AI assistance into their workflow. It takes some getting used to, but it can be worth it.
13/N