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AI-Assisted Discovery and Construction of a Counterexample to the Convergence of Three-Block ADMM with the Identity Matrix as its Third Constraint Block

Kenan Xu, Xiangfeng Wang

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 for this record

What was already known

Earlier direct three-block ADMM counterexamples used nonidentity third blocks; convergence with an identity slack block remained unresolved.

AI involvement AI-centered

Humans set the certifiable search requirements. Codex/GPT-5.6 Sol generated the period-66 rational witness; Kimi K3 independently generated the period-23 candidate. Xu and Wang explained and checked the certificate route.

Agents supplied the decisive rational periodic witnesses and certificate routes under substantial human direction; the jointly refined state reduction precludes an autonomy claim.

Importance Notable

Exact periodic witnesses answer the identity-slack convergence question negatively.

Limitations

The two cycles come from distinct searches and have different dynamical claims. Their rational certificates and raw iteration replay are reported checked by project verifiers; no separate full human proof check is documented.

Technical verification Formal (exact computation)

The existence counterexamples are reported checked by exact rational certificate verifiers, including raw ADMM replay, strict signs, closure and minimal period; AIProofWatch did not execute them.

Formal checking

exact rational certificate verification

Checker
Xu et al. project exact-rational verifier
Scope
Theorem 2.1 period-66 orbit: positive-definite data, KKT point, 66-step closure, strict projection branches, raw replay and minimal period
Basis
Reported by the linked checker

exact rational certificate and Sylvester-criterion checks

Checker
Xu et al. period-23 certificate verifier
Scope
Proposition 4.1: 23-step closure, distinct phases, strict margin and local Lyapunov attraction
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The authors explain the active-set itinerary, reduced affine maps, certificate construction and distinct dynamical scopes.

Correction and publication history