AI 中文总结
本文证明了Malitsky-Tam前向-反射-后向分裂算法步长上界$1/(2L)$的紧致性，通过构造斜旋转实例给出精确稳定性阈值，并推广到一般斜旋转族，同时提供反例验证。
AI 中文摘要
Malitsky和Tam提出的前向-反射-后向分裂算法在步长$\lambda\in(0,\tfrac{1}{2L})$时弱收敛，其中$B$是单调且$L$-Lipschitz的，该界限是否紧致的问题已被记录为开放问题\cite{GS}。我们证明该界限是紧致的：对于匹配斜对称实例$A=LJ$，$B=LJ$（$J$为$\R^2$中逆时针旋转$\pi/2$的算子），当$\lambda\ge\tfrac{1}{2L}$时迭代不收敛，当$\lambda>\tfrac{1}{2L}$时迭代发散。更一般地，对于斜旋转族$A=\gamma J$，$B=J$，精确的稳定性阈值为$\lambda^\star(\gamma)=1/\sqrt{(1+\gamma)(3-\gamma)}$，该阈值在匹配斜对称$\gamma=1$时达到最小值$1/2$，并在$\gamma=0$时恢复反射梯度常数$1/\sqrt3$。同一实例也是Cevher和Vũ提出的反射-前向-后向方法的反例，该方法在线性算子上的行为与前向-反射-后向方法一致。
英文摘要
The forward-reflected-backward splitting of Malitsky and Tam converges weakly for step sizes $λ\in(0,\tfrac{1}{2L})$, where $B$ is monotone and $L$-Lipschitz, and the question whether this bound is tight has been recorded as open \cite{GS}. We show that it is: for the matched-skew instance $A=LJ$, $B=LJ$ ($J$ the counterclockwise rotation by $π/2$ in $\R^2$), the iterates fail to converge for every $λ\ge\tfrac{1}{2L}$ and diverge for $λ>\tfrac{1}{2L}$. More generally, for the skew--rotation family $A=γJ$, $B=J$, the exact stability threshold is $λ^\star(γ)=1/\sqrt{(1+γ)(3-γ)}$, which attains its minimum $1/2$ at the matched skew $γ=1$ and recovers the reflected gradient constant $1/\sqrt3$ at $γ=0$. The same instance is a counterexample for the reflected--forward--backward method of Cevher and Vũ, which coincides with forward-reflected-backward on linear operators.
Comments5 pages, 2 figures
