"concave envelope" meaning in English

{
  "forms": [
    {
      "form": "concave envelopes",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "concave envelope (plural concave envelopes)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [
            "Entries with incorrect language header",
            "Entry maintenance"
          ],
          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Mathematics",
          "orig": "en:Mathematics",
          "parents": [
            "Formal sciences",
            "Sciences",
            "All topics",
            "Fundamental"
          ],
          "source": "w"
        }
      ],
      "coordinate_terms": [
        {
          "word": "convex envelope"
        }
      ],
      "examples": [
        {
          "ref": "1988, Ferenc Forgó, Nonconvex Programming, page 33",
          "text": "No immediate use of Theorem 4 can be made computationally since, with the exception of a few special cases (e.g. rectangular S and separable f) it is very hard to construct concave envelopes and convex hulls.",
          "type": "quotation"
        },
        {
          "ref": "2004, C. A. Meyer, C. A. Floudas, “Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave Envelopes”, in Christodoulos A. Floudas, Panos M. Pardalos, editors, Frontiers in Global Optimization, Springer,, page 327",
          "text": "Explicit expressions defining the facets of the convex and concave envelopes for trilinear monomials, with positive or negative bounded domains for each variable, are derived in this paper.",
          "type": "quotation"
        },
        {
          "ref": "2012, Joseph Geunes, Demand Flexibility in Supply Chain Planning, Springer (Kluwer Academic), page 20",
          "text": "The cost of this order plan is linear in price, and the associated line must form a segment of the piecewise linear concave envelope.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the lowest-valued concave function that overestimates or equals f over S."
      ],
      "id": "en-concave_envelope-en-noun-~q4GJ8iQ",
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "convex hull",
          "convex hull"
        ],
        [
          "concave function",
          "concave function"
        ],
        [
          "overestimate",
          "overestimate"
        ]
      ],
      "qualifier": "optimisation theory",
      "raw_glosses": [
        "(mathematics, optimisation theory, of a function on a set) For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the lowest-valued concave function that overestimates or equals f over S."
      ],
      "raw_tags": [
        "of a function on a set"
      ],
      "synonyms": [
        {
          "sense": "optimisation theory",
          "word": "upper concave envelope"
        }
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "concave envelope"
}

{
  "coordinate_terms": [
    {
      "word": "convex envelope"
    }
  ],
  "forms": [
    {
      "form": "concave envelopes",
      "tags": [
        "plural"
      ]
    }
  ],
  "head_templates": [
    {
      "args": {},
      "expansion": "concave envelope (plural concave envelopes)",
      "name": "en-noun"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
        "English countable nouns",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English nouns",
        "English terms with quotations",
        "en:Mathematics"
      ],
      "examples": [
        {
          "ref": "1988, Ferenc Forgó, Nonconvex Programming, page 33",
          "text": "No immediate use of Theorem 4 can be made computationally since, with the exception of a few special cases (e.g. rectangular S and separable f) it is very hard to construct concave envelopes and convex hulls.",
          "type": "quotation"
        },
        {
          "ref": "2004, C. A. Meyer, C. A. Floudas, “Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave Envelopes”, in Christodoulos A. Floudas, Panos M. Pardalos, editors, Frontiers in Global Optimization, Springer,, page 327",
          "text": "Explicit expressions defining the facets of the convex and concave envelopes for trilinear monomials, with positive or negative bounded domains for each variable, are derived in this paper.",
          "type": "quotation"
        },
        {
          "ref": "2012, Joseph Geunes, Demand Flexibility in Supply Chain Planning, Springer (Kluwer Academic), page 20",
          "text": "The cost of this order plan is linear in price, and the associated line must form a segment of the piecewise linear concave envelope.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the lowest-valued concave function that overestimates or equals f over S."
      ],
      "links": [
        [
          "mathematics",
          "mathematics"
        ],
        [
          "convex hull",
          "convex hull"
        ],
        [
          "concave function",
          "concave function"
        ],
        [
          "overestimate",
          "overestimate"
        ]
      ],
      "qualifier": "optimisation theory",
      "raw_glosses": [
        "(mathematics, optimisation theory, of a function on a set) For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the lowest-valued concave function that overestimates or equals f over S."
      ],
      "raw_tags": [
        "of a function on a set"
      ],
      "topics": [
        "mathematics",
        "sciences"
      ]
    }
  ],
  "synonyms": [
    {
      "sense": "optimisation theory",
      "word": "upper concave envelope"
    }
  ],
  "word": "concave envelope"
}