Convergence of Three-Block ADMM with Identity Third Block
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Statement
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.
Context
The convergence of multi-block ADMM is a well-known question in optimization since Chen-He-Ye-Yuan's celebrated 2016 counterexample, and the identity-third-block case was the natural surviving hope. Specialist but widely recognisable in the field.
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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.
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