发表机构
Lanzhou University; Peking University（兰州大学; 北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对Dai关于循环最速下降法在凸二次函数上R-超线性收敛的普遍断言，本文构造两个闭式反例（二维平衡锯齿轨道和三维周期二轨道），证明该断言在阈值m≥⌈(n+1)/2⌉下不成立。
AI 中文摘要
循环最速下降法（CSD）在每个循环中重新计算一次精确的最速下降步长，并将其重复用于m次更新。Dai在ICM 2022综述中提出，当m≥⌈(n+1)/2⌉时，CSD在n维凸二次函数上可能R-超线性收敛。我们通过两个在所述阈值下的闭式轨道否证了这一断言的普遍形式。首先，对于n=m=2，A=diag(1,3)，b=0，x₀=(1,1/3)ᵀ，该方法遵循非终止的平衡锯齿轨道x_k=2^{-k}(1,(-1)^k/3)ᵀ，其连续误差范数之比为1/2。其次，对于n=3，m=2，A=diag(1,2,3)，我们展示了一个全支撑、非共振的循环边界射影周期二轨道，满足g_{k+4}=g_k/49和x_{k+4}=x_k/49。第二个构造是真正三维的，并非二维锯齿。因此，普遍性论断通过经典的平衡锯齿机制和一种不同的非锯齿周期二机制均告失败。
英文摘要
Cyclic steepest descent (CSD) recomputes the exact steepest-descent stepsize once per cycle and reuses it for $m$ updates. Dai's ICM 2022 survey describes CSD as likely to converge $R$-superlinearly on $n$-dimensional convex quadratics when $m\ge\lceil(n+1)/2\rceil$. We disprove the universal form of this assertion by two closed-form orbits at the stated threshold. First, for $n=m=2$, $A=\operatorname{diag}(1,3)$, $b=0$, and $x_0=(1,1/3)^{\mathsf{T}}$, the method follows the nonterminating balanced-zigzag orbit $x_k=2^{-k}(1,(-1)^k/3)^{\mathsf{T}}$, whose successive error norms have ratio $1/2$. Second, for $n=3$, $m=2$, and $A=\operatorname{diag}(1,2,3)$, we exhibit a full-support, nonresonant cycle-boundary projective period-two orbit with $g_{k+4}=g_k/49$ and $x_{k+4}=x_k/49$. This second construction is genuinely three-dimensional and is not a two-dimensional zigzag. Thus the universal claim fails through both the classical balanced-zigzag mechanism and a distinct non-zigzag period-two mechanism.
Comments6 pages. An earlier version (Zenodo v3) was published on 31 August 2026: https://zenodo.org/records/22209278 . The exposition has been revised; the main conclusions are unchanged
