concave envelope

concave envelope (plural concave envelopes)

1. (mathematics, optimisation theory, of a function on a set) For a given set ${\displaystyle S\subseteq \mathbb {R} ^{n}}$ and real-valued function f defined on the convex hull conv(S), the lowest-valued concave function that overestimates or equals f over S.
   • 1988, Ferenc Forgó, Nonconvex Programming, page 33,

     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.

   • 2004, C. A. Meyer, C. A. Floudas, Trilinear Monomials with Positive or Negative Domains: Facets of the Convex and Concave Envelopes, Christodoulos A. Floudas, Panos M. Pardalos (editors), Frontiers in Global Optimization, Springer, page 327,

     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.

   • 2012, Joseph Geunes, Demand Flexibility in Supply Chain Planning, Springer (Kluwer Academic), page 20,

     The cost of this order plan is linear in price, and the associated line must form a segment of the piecewise linear concave envelope.

• (optimisation theory): upper concave envelope

• convex envelope