Convergence of direct three-block ADMM with identity constraints
Follow the latest progress by email.
Does direct three-block ADMM converge for strongly convex quadratic programs when the third constraint block is an identity slack block? General multi-block divergence alone does not settle this restricted case.
Why it matters and what counts as progress
Why it matters: Identity slack is common in reformulations, so its convergence behavior is a distinct boundary of the method.
What counts as progress: A convergence theorem or an exact bounded non-KKT periodic orbit under the identity-block hypotheses, with any attraction claim proved separately.
How the problem developed
- 2016 · Counterexample [1]
Changhong Chen, Bingsheng He, Yinyu Ye and X. Yuan Showed that the direct extension of ADMM to multi-block convex minimization need not converge.
- 14 Aug 2026 · Counterexample [1]
Kenan Xu and Xiangfeng Wang with AI assistance Gave an exact rational three-block counterexample with identity third constraint block and a bounded nonconvergent orbit of period 66.
AI contribution
AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as its Third Constraint Block
Xu and Wang exhibit a strongly convex rational quadratic program with A=B=I₂ and identity third block whose direct three-block ADMM has a bounded non-KKT orbit of minimal period 66. A separate Kimi search gives a locally attracting period-23 orbit in dimension three that can be converted to all-identity blocks.
Sources
Website timeline
Publication dates and editorial changes.
AI-found exact cycles refute identity-slack three-block ADMM convergence