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∈(0,1/2L), 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 R2), the iterates fail to converge for every λ≥1/2L and diverge for λ>1/2L. More generally, for the skew--rotation family A=γJ, B=J, the exact stability threshold is λ⋆(γ)=1/√((1+γ)(3−γ)), which attains its minimum 1/2 at the matched skew γ=1 and recovers the reflected gradient constant 1/√3 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.
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BibTeX
@misc{shehu2026sharp,
title = {{The sharp step-size constant for one-call reflection splittings on monotone inclusions}},
author = {Yekini Shehu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18373},
eprint = {2609.18373},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18373},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The sharp step-size constant for one-call reflection splittings on monotone inclusions
AU - Yekini Shehu
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18373
UR - https://arxiv.org/abs/2609.18373
AB - The forward-reflected-backward splitting of Malitsky and Tam converges weakly for step sizes λ∈(0,1/2L), 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≥1/2L and diverge for λ>1/2L. More generally, for the skew--rotation family A=γJ, B=J, the exact stability threshold is λ^⋆(γ)=1/√((1+γ)(3−γ)), which attains its minimum 1/2 at the matched skew γ=1 and recovers the reflected gradient constant 1/√3 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.
ER -