convex envelopes

Noun

convex envelope (plural convex envelopes)

1. Convex hull.
   • 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.
2. (optimisation theory) 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.
   • 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...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