Convex envelopes for edge-concave functions

Abstract

Deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions. In this paper we describe the structure of the polyhedral convex envelopes of edge-concave functions over polyhedral domains using geometric arguments. An algorithm for computing the facets of the convex envelope over hyperrectangles in ℝ3 is described. Sufficient conditions are described under which the convex envelope of a sum of edge-concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions.