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  "title": "Cocoercivity does not give a quadratic residual rate for Douglas–Rachford splitting",
  "shortTitle": "An exact limit to a proposed splitting guarantee",
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  "oneLine": "Small exact counterexamples refute a proposed squared-residual guarantee; the correct uniform worst-case order is inverse-linear, with carefully restricted scalar-step consequences.",
  "abstract": "Douglas–Rachford splitting tackles a coupled problem through simpler resolvent calculations. A conjecture proposed that cocoercivity of one operator, together with a convex subdifferential for the other, would give a uniform inverse-square bound on the squared update residual. This candidate refutes that statement using exact scalar and planar examples and proves that the correct uniform worst-case squared-residual order is inverse-linear for every fixed admissible parameter triple. Hinge-and-rotation witnesses also exclude an elementary finite-horizon repair considered in the paper. Additional results describe prescribed linear schedules and adaptive scalar-policy limits using credited classical and recent ingredients. An elementary corollary gives the exact linear-subclass infimum for scalars in the conventional open interval, without a nonlinear upper claim. These are fixed-point residual results, not objective-gap bounds or evidence of practical solver failure. The source paper's separate objective acceleration conjectures remain open. A bounded search located no earlier direct refutation; inaccessible full text leaves overlap of the scheduling synthesis unresolved.",
  "datePublished": "2026-11-05",
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    "Exact scalar and planar witnesses refute Conjecture 3.5 of Abbaszadehpeivasti and Zamani (2025), version 2, within its stated assumptions.",
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    "For linear operators and c>2, scalars restricted to (0,2) have an exact infimum approached but not attained at the boundary; no matching nonlinear upper bound is claimed.",
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    {
      "citation": "Abbaszadehpeivasti and Zamani (2025). On the convergence rate of the Douglas–Rachford splitting algorithm. arXiv, version 2. Source of Conjecture 3.5.",
      "url": "https://arxiv.org/abs/2509.06676v2",
      "doi": "10.48550/arXiv.2509.06676"
    },
    {
      "citation": "Giselsson (2017). Tight global linear convergence rate bounds for Douglas–Rachford splitting. Journal of Fixed Point Theory and Applications 19, 2241–2270.",
      "url": "https://doi.org/10.1007/s11784-017-0417-1",
      "doi": "10.1007/s11784-017-0417-1"
    },
    {
      "citation": "Gu and Yang (2019). Optimal nonergodic sublinear convergence rate of proximal point algorithm for maximal monotone inclusion problems. arXiv, version 2. Inherited sharp upper bound.",
      "url": "https://arxiv.org/abs/1904.05495v2",
      "doi": "10.48550/arXiv.1904.05495"
    },
    {
      "citation": "Lieder (2021). On the convergence rate of the Halpern-iteration. Optimization Letters 15, 405–418.",
      "url": "https://doi.org/10.1007/s11590-020-01617-9",
      "doi": "10.1007/s11590-020-01617-9"
    },
    {
      "citation": "Contreras and Cominetti (2022). Optimal error bounds for nonexpansive fixed-point iterations in normed spaces. Author manuscript, version 3.",
      "url": "https://arxiv.org/abs/2108.10969v3",
      "doi": "10.48550/arXiv.2108.10969"
    },
    {
      "citation": "He, Dong, Tian and Li (2021). On the optimal relaxation parameters of Krasnosel'ski–Mann iteration. Optimization 70(9), 1959–1986. Abstract inspected; full-text overlap remains unresolved.",
      "url": "https://doi.org/10.1080/02331934.2020.1767101",
      "doi": "10.1080/02331934.2020.1767101"
    },
    {
      "citation": "Sarkar and Aggarwal (2026). Non-expansive two-time-scale stochastic approximation: A fixed-schedule one-quarter barrier and bias-corrected acceleration. arXiv. Published rotation-product antecedent.",
      "url": "https://arxiv.org/abs/2607.13414",
      "doi": "10.48550/arXiv.2607.13414"
    },
    {
      "citation": "Hartz (2025). On von Neumann's inequality on the polydisc. Mathematische Annalen 391, 5235–5264.",
      "url": "https://doi.org/10.1007/s00208-024-03040-2",
      "doi": "10.1007/s00208-024-03040-2"
    },
    {
      "citation": "von Neumann (1950). Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Erhard Schmidt zum 75. Geburtstag in Verehrung gewidmet. Mathematische Nachrichten 4(1–6), 258–281. Registry title includes the dedication; the manuscript uses the shorter title. Classical inequality inspected through Hartz.",
      "url": "https://doi.org/10.1002/mana.3210040124",
      "doi": "10.1002/mana.3210040124"
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