The sharp step-size constant for one-call reflection splittings on monotone inclusions

Abstract: 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 $\R<sup>2$), the iterates fail to converge for every $λ\ge\tfrac{1}{2L}$ and diverge for $λ&gt;\tfrac{1}{2L}$. More generally, for the skew--rotation family $A=γJ$, $B=J$, the exact stability threshold is $λ<sup>\star(γ)=1/\sqrt{(1+γ)(3-γ)}$, 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.