concave envelope

plural concave envelopes

1. optimisation theorymathematicsFor a given set S⊆ℝⁿ and real-valued function f defined on the convex hull conv(S), the lowest-valued concave function that overestimates or equals f over S.

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

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

   2 more examples

   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.

   1988, Ferenc Forgó, Nonconvex Programming, page 33

   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.

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

   Synonyms: upper concave envelope