{
  "abstract": "Condat's algorithm is an efficient dynamic-threshold method for projection onto the simplex, but its extension to weighted equality constraints and the algorithmic roles of resetting and removal have received limited analysis. We develop a dynamic-threshold algorithm (DTA) for a continuous quadratic knapsack problem with a weighted equality constraint. DTA maintains a threshold invariant through three operations--addition, reset, and removal--and we establish its finite termination and correctness. A sufficient condition under which reset cannot occur motivates a simpler no-reset variant, NDTA. We construct instances for which DTA runs in $Θ (n)$ time whereas NDTA requires $Θ(n^2)$ time, although both algorithms have quadratic worst-case complexity. We further show that, when the weight ratio and the number of deletions per removal pass are bounded, a linear number of passes with positive threshold increments requires the minimum nonzero gap between input values, normalized by the data range, to be at most $\\exp [- Θ(n \\log n)]$. Numerical experiments with up to 10^7 variables demonstrate that DTA and NDTA achieve approximately linear empirical scaling, and outperform Secant, WMVA, Variable Fixing, Newton, Median Search, Heap, and Sort in running time.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.00740",
  "author_total": 3,
  "authors": [
    {
      "name": "Yong-Jin Liu"
    },
    {
      "name": "Peicheng Xie"
    },
    {
      "name": "Chuan Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fj92bmrf35mvv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.00740",
  "fulltext_url": null,
  "id": "fj92bmrf35mvv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.00740",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.OC"
    ],
    [
      "openalex",
      "2612|Advanced Optimization Algorithms Research"
    ]
  ],
  "openalex": "W7172404945",
  "pdf_url": "https://arxiv.org/pdf/2608.00740",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fj92bmrf35mvv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.00740",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.00740"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OC",
      "subfield": "optimisation-control"
    }
  ],
  "title": "Dynamic-Threshold Algorithms for the Continuous Quadratic Knapsack Problem: Reset Mechanisms and Complexity",
  "title_text": "Dynamic-Threshold Algorithms for the Continuous Quadratic Knapsack Problem: Reset Mechanisms and Complexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/fj92bmrf35mvv",
  "venue": "arXiv",
  "versions": []
}