{
  "abstract": "We study how to compute approximate solutions of inconsistent systems of max-min fuzzy relational equations with respect to the $L_1$ and $L_2$ norms using mixed-integer programming. We assign one error bound to each equation and collect these bounds into a vector. We call this error-bound vector admissible if there exists a right-hand side vector that defines a consistent max-min system with the original matrix and satisfies the prescribed componentwise error bounds relative to the original right-hand side vector. We characterize admissibility and, for every admissible error-bound vector, we explicitly construct such a right-hand side vector and a solution of the resulting consistent system. We also identify, for each equation, column indices that can be discarded from the admissibility criterion. This allows us to reformulate the $L_1$ and $L_2$ approximation problems as equivalent optimization problems over the set of admissible error-bound vectors. We show that the set of admissible error-bound vectors is a finite union of non-empty compact polytopes, and we describe their structure. We then derive an exact mixed-integer linear representation of this set, yielding an exact mixed-integer linear program (MILP) for the $L_1$ problem and an exact mixed-integer quadratic program (MIQP) with a convex quadratic objective for the $L_2$ problem. The results extend to min-max systems, and computational experiments on synthetic data show improved performance of the proposed MIQP formulation compared with two existing formulations on the tested instances.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Ismaïl Baaj"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3mpwzk284y1fr.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "3mpwzk284y1fr",
  "issns": [],
  "landing_url": "https://hal.science/hal-05847442v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.info"
    ],
    [
      "hal",
      "0.stat"
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  "openalex": null,
  "pdf_url": "https://hal.science/hal-05847442/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3mpwzk284y1fr.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:20:44Z",
      "native_id": "hal-05847442",
      "source": "hal",
      "url": "https://hal.science/hal-05847442v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.stat",
      "subfield": "statistics"
    }
  ],
  "title": "Computing Approximate Solutions of Inconsistent Max-Min Fuzzy Relational Equations by Mixed-Integer Programming",
  "title_text": "Computing Approximate Solutions of Inconsistent Max-Min Fuzzy Relational Equations by Mixed-Integer Programming",
  "type": "preprint",
  "updated_at": "2026-11-03T12:20:17Z",
  "url": "https://research.draconitic.com/paper/3mpwzk284y1fr",
  "venue": "HAL",
  "versions": []
}