AI-Assisted Counterexample to Three-Block ADMM Convergence with Identity Constraint
Utilizing artificial intelligence, researchers have demonstrated a counterexample indicating that the three-block alternating direction method of multipliers (ADMM) might not converge, even when the third constraint block is represented by the identity matrix. This addresses a previously unanswered question in the field of optimization theory. The counterexample was identified through Codex with GPT-5.6 Sol, an AI programming assistant, and was confirmed via a piecewise-affine reduction approach. The findings, published on arXiv (2608.14396), reveal that direct three-block ADMM can fail, despite the first two blocks being strongly convex quadratics. This is particularly noteworthy as two-block ADMM has established convergence assurances, while the extension to three blocks has been problematic, especially concerning the identity third block. The research underscores AI's increasing influence in mathematical advancements.
Key facts
- Three-block ADMM may fail to converge when the third constraint block is the identity matrix.
- The counterexample was constructed using Codex with GPT-5.6 Sol.
- The counterexample is explicit and rational.
- Verification was done along a piecewise-affine reduction path.
- The first two blocks are strongly convex quadratics.
- The paper is available on arXiv with ID 2608.14396.
- This resolves an open question in optimization theory.
- The finding extends existing counterexamples for three-block ADMM.
Entities
Institutions
- arXiv