Convergence of Three-Block ADMM with Identity Third Block
After Chen-He-Ye-Yuan's counterexample to direct three-block ADMM, the subclass in which the third constraint block is the identity matrix remained unresolved: the literature contained neither a convergence proof nor a counterexample. Disproved: an explicit rational counterexample exists in which the first two blocks are strongly convex quadratics and direct three-block ADMM produces a bounded nonconvergent orbit of period 66, verified by exact checks along a piecewise-affine reduction path.
- Result
- Disproved
- Status
- Resolved
- AI contribution
- AI co-developed
- Method
- Construction
- Field
- Optimization - splitting methods
- Posed by
- Open subclass left by Chen, He, Ye, Yuan (2016)
- Year posed
- 2016
- Years open
- 10y
- Solved
- 2026-08-14
- Model
- GPT-5.6 Sol (Codex)
- Vendor
- OpenAI
- Collaborators
- Kenan Xu, Xiangfeng Wang
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 15 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
The paper is framed as AI-assisted discovery in its own title: using Codex with GPT-5.6 Sol, the authors construct the explicit rational counterexample candidate and verify it along a piecewise-affine reduction path, with exact checks establishing the period-66 nonconvergent orbit; the same Codex workflow guides a further study.
Verification
Checked by this site on 17 August 2026 against the paper's LaTeX (arXiv:2608.14396): the open status of the identity-third-block subclass is stated in the abstract and introduction, and the construction-and-verification account is as quoted. The exact checks were not re-run here. Days-old preprint, no independent review.