发表机构
School of Mathematical Sciences, East China Normal University（华东师范大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究借助AI工具构造出以单位矩阵为第三约束块的三块ADMM的收敛反例，还分析了乘子松弛对收敛性的影响，对比了不同AI工具在数学研究中的表现。
AI 中文摘要
交替方向乘子法（ADMM）作为一种里程碑式算法，在过去二十年中吸引了大量研究关注并得到广泛实际应用。众所周知，尽管双块ADMM具有完善的理论收敛保证，但其直接扩展到三块情形可能无法收敛，现有反例[5]已证明这一点。然而据我们所知，当第三个约束块为单位矩阵时，该子类问题尚未得到解决：现有文献既未给出一般收敛性证明，也未提供反例。本文给出否定答案：即使前两个块为强凸二次型，直接三块ADMM也可能不收敛。我们使用Codex结合GPT-5.6 Sol，构造了一个显式有理反例候选，并沿分段仿射约化路径对其进行验证；精确检查表明，对该实例的直接三块ADMM会产生周期为66的有界非收敛轨道。在同一Codex工作流中，我们进一步指导研究乘子松弛，并阐明在固定实例和类级别上何时可恢复收敛：依赖于问题的小对偶步可恢复收敛，而正相对步无法在整个类上统一生效。此外，我们还测试了最新的Kimi Code工具，使用Kimi K3模型，未借助Codex候选或特定项目路线指导；其沿不同路径生成了一个精确的局部吸引周期23的证明，可转换为等价的全单位矩阵实例。该比较表明，不同的研究利用配置会影响所探索的数学对象和所追求的证明。
英文摘要
The alternating direction method of multipliers (ADMM), as a landmark algorithm, has attracted tremendous research attention and extensive practical applications over the past two decades. It is well known that, although the two-block ADMM enjoys well-established theoretical convergence guarantees, its direct extension to the three-block case may fail to converge, as demonstrated by existing counterexamples [5]. However, to the best of our knowledge, the case in which the third constraint block is the identity remains unresolved: the existing literature gives neither a general convergence proof nor a counterexample for this subclass. In this paper, we give a negative answer: direct three-block ADMM may fail even when the first two blocks are strongly convex quadratics. Using Codex with GPT-5.6 Sol, we construct an explicit rational counterexample candidate and verify it along a piecewise-affine reduction path; exact checks show that direct three-block ADMM on this instance produces a bounded nonconvergent orbit of period 66. Within the same Codex workflow, we further guide a study of multiplier relaxation and clarify when convergence can be restored at the fixed-instance and class levels: a problem-dependent small dual step can restore convergence, whereas no positive relative step works uniformly over the whole class. Furthermore, we also test the recent Kimi Code with Kimi K3 model without the Codex candidate or project-specific route guidance; along a different path it produces an exact locally attracting period-23 certificate, convertible to an equivalent all-identity instance. The comparison suggests that different research-harness configurations can shape the mathematical objects explored and the certificates pursued.
Comments34 pages
