AI-Assisted Counterexamples Show Three-Block ADMM Can Cycle with an Identity Third Block
Kenan Xu and Xiangfeng Wang give exact periodic counterexamples showing that direct three-block ADMM can fail to converge even with an identity third constraint block and strongly convex quadratic first blocks; Codex with GPT-5.6 Sol and Kimi Code with Kimi K3 produced distinct certified routes, while the paper also limits what multiplier relaxation can guarantee.
In a preprint first uploaded on August 14, 2026, Kenan Xu and Xiangfeng Wang give a negative answer to the identity-slack convergence question for direct three-block ADMM. They construct a convex quadratic instance with an identity third constraint block for which the algorithm has a bounded, nonconvergent orbit, showing that this structural restriction alone does not ensure convergence.
The paper gives an exact rational period-66 counterexample with strongly convex first and second blocks. A Codex workflow using GPT-5.6 Sol reportedly contributed to the representation, candidate construction, exact certification, and subsequent analysis of multiplier relaxation. A separate blind route using Kimi Code with Kimi K3 produced an exact locally attracting period-23 certificate that can be converted to an all-identity instance. The authors report that exact certificates, rather than floating-point behavior, support the theorem-level claims.
The same study finds that a sufficiently small, problem-dependent dual step can restore convergence on a fixed instance, while no positive relative step works uniformly over the entire class. The companion repository provides rational data, verification programs, prompt records, and provenance materials for reproducing the two periodic certificates.