A Convex Envelope Formula for Multilinear Functions

Authors:
Anatoliy D. Rikun
Affiliations:
East Coast Product Group, 45 Winnett St., Hamden, CT 06517, U.S.A. (email: rikun@ptc.com)
Venue:
Journal of Global Optimization
Year:
1997

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Abstract

Convex envelopes of multilinear functions on a unit hypercube arepolyhedral. This well-known fact makes the convex envelopeapproximation very useful in the linearization of non-linear 0–1programming problems and in global bilinear optimization. This paperpresents necessary and sufficient conditions for a convex envelope to be apolyhedral function and illustrates how these conditions may be used inconstructing of convex envelopes. The main result of the paper is a simpleanalytical formula, which defines some faces of the convex envelope of amultilinear function. This formula proves to be a generalization of the wellknown convex envelope formula for multilinear monomial functions.