convex envelope

Noun

of a set of a function on a set

(mathematical analysis, of a set) Convex hull.

Japanese Meaning

凸包

What is this buttons?

The convex envelope of a set of points is the smallest convex set that contains them.

Google Translation / DeepL Translation

点集合の凸包は、それらを含む最小の凸集合です。

Details

Sense（1）

(mathematical analysis, of a set) Convex hull.

In E the closed convex envelope of a compact (resp. weakly compact) set is 𝜏(E,E')-complete.

Google Translation / DeepL Translation

Details

Sense（2）

(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.

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).

Google Translation / DeepL Translation