{
  "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 $\\R^2$), 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18373",
  "author_total": 1,
  "authors": [
    {
      "name": "Yekini Shehu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dms37h9cgbavw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18373",
  "fulltext_url": null,
  "id": "dms37h9cgbavw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18373",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "1703|Optimization and Variational Analysis"
    ]
  ],
  "openalex": "W7213507547",
  "pdf_url": "https://arxiv.org/pdf/2609.18373",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dms37h9cgbavw.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18373",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18373"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "The sharp step-size constant for one-call reflection splittings on monotone inclusions",
  "title_text": "The sharp step-size constant for one-call reflection splittings on monotone inclusions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/dms37h9cgbavw",
  "venue": "arXiv",
  "versions": []
}