{
  "abstract": "A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective function subject to multiple two-sided linear inequalities intersected with a low-rank and spectral constrained domain. Although solving LSOP is generally NP-hard, its partial convexification (i.e., replacing the domain with its convex hull), termed “LSOP-R,\" is often tractable and yields a high-quality solution. This motivates us to study the strength of LSOP-R. Specifically, we derive rank bounds for any extreme point of LSOP-R in different matrix spaces and prove their tightness. The proposed rank bounds recover two well-known results in the literature from a fresh angle and allow us to derive sufficient conditions under which the relaxation LSOP-R is equivalent to LSOP. To effectively solve LSOP-R, we develop a column generation algorithm with a vector-based convex pricing oracle and a rank-reduction algorithm, which ensures that the output solution always satisfies the theoretical rank bound. Finally, we numerically verify the strength of LSOP-R and the efficacy of the proposed algorithms.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Yongchun Li"
    },
    {
      "name": "Weijun Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bg7qar9tmfn8c.bib",
  "cited_by": 0,
  "doi": "10.1007/s10107-025-02316-y",
  "fulltext_url": null,
  "id": "bg7qar9tmfn8c",
  "issns": [
    "0025-5610",
    "1436-4646"
  ],
  "landing_url": "https://hal.science/hal-05783679v1",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-oc"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7116797934",
  "pdf_url": "https://link.springer.com/content/pdf/10.1007/s10107-025-02316-y.pdf",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": "Springer Verlag",
  "ris": "https://research.draconitic.com/paper/bg7qar9tmfn8c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:50Z",
      "native_id": "hal-05783679",
      "source": "hal",
      "url": "https://hal.science/hal-05783679v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:1.math.math-oc",
      "subfield": "optimisation-control"
    }
  ],
  "title": "On the partial convexification for low-rank spectral optimization: rank bounds and algorithms",
  "title_text": "On the partial convexification for low-rank spectral optimization: rank bounds and algorithms",
  "type": "peer-reviewed",
  "updated_at": "2026-10-30T12:17:50Z",
  "url": "https://research.draconitic.com/paper/bg7qar9tmfn8c",
  "venue": "Mathematical Programming",
  "versions": []
}