{
  "abstract": "This paper continues our work on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences. We extend the method developed for the Schmüdgen-type hierarchy to the Putinar-type, Krivine--Stengle-type, extended-Handelman-type, and Hol-Scherer-type hierarchies. The main idea is to lift a pseudo-moment sequence to a simple set, approximate it by a moment sequence, and project the atoms of a representing measure onto the original feasible set. The Łojasiewicz inequality then converts the error in the defining constraints into an error in the moments. With the Łojasiewicz exponent $0<L\\leq 1$, we obtain the convergence rates $\\mathrm{O}((\\log_2 r)^{3L/2}/r^{L})$ for the Putinar-type hierarchy and $\\mathrm{O}(1/r^{L/2})$ for the normalized Krivine--Stengle-type and extended-Handelman-type hierarchies. In the matrix setting, we utilize a Chebyshev-type kernel on $[-1,1]^n$ to derive the rate $\\mathrm{O}((\\log_2 r)^{3L/2}/r^{L})$ for the Hol-Scherer-type hierarchy. Together with our preceding work, the results provide a universal method for studying convergence rates of different types of moment-SOS hierarchies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37286",
  "author_total": 2,
  "authors": [
    {
      "name": "Hoang Anh Tran"
    },
    {
      "name": "Kim-Chuan Toh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/66adsjccfv1z4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37286",
  "fulltext_url": null,
  "id": "66adsjccfv1z4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37286",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7214939014",
  "pdf_url": "https://arxiv.org/pdf/2609.37286",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/66adsjccfv1z4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37286",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37286"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences",
  "title_text": "Extension on the convergence rates of moment-SOS hierarchies via approximation of truncated moment sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/66adsjccfv1z4",
  "venue": "arXiv",
  "versions": []
}