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  "title": "Transient objective convergence of spectrally weighted gradient descent",
  "shortTitle": "Two steps, two different yardsticks",
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  "oneLine": "Three spectral modes defeat a proposed two-factor objective bound, while a sharp transient theorem separates growing condition numbers from fixed-problem long-time rates.",
  "abstract": "This unrefereed AI-assisted candidate studies ordinary objective gaps for established weighted-gradient methods on positive-definite quadratics. Exact examples distinguish one-factor and two-factor readings of a two-step question. Fixed-horizon asymptotics and controlled growing horizons quantify transient behavior without refuting proved convergence results or claiming fixed-accuracy iteration lower bounds.",
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    "quadratic optimization",
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    "Fixed-horizon worst-case objective decrease has leading coefficient 2 omega K as condition number grows.",
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    {
      "citation": "Pronzato, Wynn and Zhigljavsky (2006). Asymptotic behaviour of a family of gradient algorithms in R^d and Hilbert spaces. Mathematical Programming.",
      "url": "https://doi.org/10.1007/s10107-005-0602-7",
      "doi": "10.1007/s10107-005-0602-7"
    },
    {
      "citation": "Huang, Dai, Liu and Zhang (2022). On the asymptotic convergence and acceleration of gradient methods. Journal of Scientific Computing.",
      "url": "https://doi.org/10.1007/s10915-021-01685-8",
      "doi": "10.1007/s10915-021-01685-8"
    },
    {
      "citation": "He (2025). The worst-case root-convergence factor of GMRES(1). arXiv v1.",
      "url": "https://arxiv.org/abs/2501.10248v1",
      "doi": "10.48550/arXiv.2501.10248"
    },
    {
      "citation": "MacDonald, Murray and Tappenden (2025). A Flexible Algorithmic Framework for Strictly Convex Quadratic Minimization. arXiv v1.",
      "url": "https://arxiv.org/abs/2505.04047v1",
      "doi": "10.48550/arXiv.2505.04047"
    },
    {
      "citation": "MacDonald, Murray and Tappenden (2025). On a family of relaxed gradient descent methods for strictly convex quadratic minimization.",
      "url": "https://doi.org/10.1007/s10589-025-00670-3",
      "doi": "10.1007/s10589-025-00670-3"
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