"convex envelope" meaning in English

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          "word": "concave envelope"
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          "text": "1965 [Holt Rinehart & Winston], Robert E. Edwards, Functional Analysis: Theory and Applications, Dover, 1995, Unabridged Corrected Edition, page 561,\nIn E the closed convex envelope of a compact (resp. weakly compact) set is τ(E,E')-complete."
        },
        {
          "text": "1987, H. G. Eggleston, S. Madan (translators), Nicolas Bourbaki, Topological Vector Spaces: Chapters 1–5, [1981, N. Bourbaki, Espaces Vectoriels Topologiques], Springer, page IR-10,\nCorollary 1. — The convex envelope of a subset A of E is identical with the set of linear combinations ∑ᵢλᵢx_i, where (x_i) is any finite family of points in A, the numbers λᵢ>0 for all i and ∑ᵢλᵢ=1."
        },
        {
          "ref": "2010, Marcel Berger, Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry, Springer, page 505",
          "text": "The polytopes are, by definition, the convex envelopes of finite sets of points of an affine space.",
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        }
      ],
      "glosses": [
        "Convex hull."
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      "id": "en-convex_envelope-en-noun-6pFqam6j",
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        "(mathematical analysis, of a set) Convex hull."
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          "word": "lower convex envelope"
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          "ref": "2004, Fabio Tardella, “On the existence of polyhedral convex envelopes”, in Christodoulos A. Floudas, Panos M. Pardalos, editors, Frontiers in Global Optimization, Springer (Kluwer Academic), page 564",
          "text": "One of the main reasons for the interest in convex envelopes is the fact that the set of global minimum points of f on S is contained in the set of global minimum points of conv_S(f) on S and the two minimum values coincide (see, e.g., [11, 17]). Hence, if the convex envelope were efficiently computable or available in closed form, one could replace the nonconvex problem of minimizing f on S with the convex problem of minimizing f on conv_S(f).",
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          "ref": "2004, C. A. Floudas, I. G. Akrotiriankis, S. Caratzoulas, C. A. Meyer, J. Kallrath, “Global Optimization in the 21st Century: Advances and Challenges”, in Ana Paula Barbosa-Póvoa, Henrique Matos, editors, European Symposium on Computer Aided Process Engineering-14: 37th European Symposium of the Working Party, Elsevier, page 25",
          "text": "Tawarmalani and Sahinidis (2001) developed the convex envelope and concave envelope for x/y over a unit hypercube, compared it to the convex relaxation proposed by Zamora and Grosmmann (1998a), (1998b), (1999), proposed a semidefinite relaxation of x/y, and suggested convex envelopes for functions of the form f(x)y² and f(x)/y.",
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        {
          "ref": "2005, Nguyen Van Thoai, “4: General Quadratic Programing”, in Charles Audet, Pierre Hansen, Giles Savard, editors, Essays and Surveys in Global Optimization, Springer, page 120",
          "text": "The concept of convex envelopes of nonconvex functions is a basic tool in theory and algorithms of global optimization, see e.g., Falk and Hoffman (1976), Horst and Tuy (1996), Horst et al. (2000).",
          "type": "quotation"
        },
        {
          "ref": "2012, Petro Bilotti, Sonia Cafieri, Jon Lee, Leo Liberti, Andrew J. Miller, On the Composition of Convex Envelopes for Quadrilinear Terms",
          "text": "Comparing the use of convex envelopes for bilinear and trilinear forms in building convex approximations for MINLPs motivated the study in [6], and comparisons involving more general functional forms motivate the present article.",
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        "For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the highest-valued convex function that underestimates or equals f over S."
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      "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 highest-valued convex function that underestimates or equals f over S."
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  "word": "convex envelope"
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          "text": "1965 [Holt Rinehart & Winston], Robert E. Edwards, Functional Analysis: Theory and Applications, Dover, 1995, Unabridged Corrected Edition, page 561,\nIn E the closed convex envelope of a compact (resp. weakly compact) set is τ(E,E')-complete."
        },
        {
          "text": "1987, H. G. Eggleston, S. Madan (translators), Nicolas Bourbaki, Topological Vector Spaces: Chapters 1–5, [1981, N. Bourbaki, Espaces Vectoriels Topologiques], Springer, page IR-10,\nCorollary 1. — The convex envelope of a subset A of E is identical with the set of linear combinations ∑ᵢλᵢx_i, where (x_i) is any finite family of points in A, the numbers λᵢ>0 for all i and ∑ᵢλᵢ=1."
        },
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          "ref": "2010, Marcel Berger, Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry, Springer, page 505",
          "text": "The polytopes are, by definition, the convex envelopes of finite sets of points of an affine space.",
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      "glosses": [
        "Convex hull."
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          "text": "One of the main reasons for the interest in convex envelopes is the fact that the set of global minimum points of f on S is contained in the set of global minimum points of conv_S(f) on S and the two minimum values coincide (see, e.g., [11, 17]). Hence, if the convex envelope were efficiently computable or available in closed form, one could replace the nonconvex problem of minimizing f on S with the convex problem of minimizing f on conv_S(f).",
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          "text": "Tawarmalani and Sahinidis (2001) developed the convex envelope and concave envelope for x/y over a unit hypercube, compared it to the convex relaxation proposed by Zamora and Grosmmann (1998a), (1998b), (1999), proposed a semidefinite relaxation of x/y, and suggested convex envelopes for functions of the form f(x)y² and f(x)/y.",
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          "ref": "2005, Nguyen Van Thoai, “4: General Quadratic Programing”, in Charles Audet, Pierre Hansen, Giles Savard, editors, Essays and Surveys in Global Optimization, Springer, page 120",
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        },
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          "ref": "2012, Petro Bilotti, Sonia Cafieri, Jon Lee, Leo Liberti, Andrew J. Miller, On the Composition of Convex Envelopes for Quadrilinear Terms",
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      "glosses": [
        "For a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the highest-valued convex function that underestimates or equals f over S."
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          "underestimate",
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      "qualifier": "optimisation theory",
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        "(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 highest-valued convex function that underestimates or equals f over S."
      ],
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      "word": "lower convex envelope"
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  "word": "convex envelope"
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