convex envelope
Language: English
Part of speech: noun
Forms: form: convex envelopes
Senses: examples: text: 1965 [Holt Rinehart & Winston], Robert E. Edwards, Functional Analysis: Theory and Applications, Dover, 1995, Unabridged Corrected Edition, page 561,
In 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,
Corollary 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.
text: The polytopes are, by definition, the convex envelopes of finite sets of points of an affine space.
ref: 2010, Marcel Berger, Geometry Revealed: A Jacob's Ladder to Modern Higher Geometry, Springer, page 505:
topics: mathematical-analysis
mathematics
sciences
glosses: Convex hull.
raw_tags: of a set
examples: 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).
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: 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.
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: 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).
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: 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.
ref: 2012, Petro Bilotti, Sonia Cafieri, Jon Lee, Leo Liberti, Andrew J. Miller, On the Composition of Convex Envelopes for Quadrilinear Terms:
topics: mathematics
sciences
qualifier: optimisation theory
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.
raw_tags: of a function on a set
Synonyms: sense: optimisation theory
word: lower convex envelope
Coordinate terms: concave envelope
upper concave envelope