Point Convergence of Nesterov's Accelerated Gradient Method
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Statement
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.
Context
The method is world-famous and the point-convergence question was known across the optimization community for four decades, with Attouch-school partial results; invisible outside the field.
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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.
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