Convex extensions and envelopes of lower semi-continuous functions |
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Authors: | Mohit Tawarmalani Nikolaos V Sahinidis |
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Institution: | (1) Krannert School of Management, Purdue University. e-mail: mtawarma@mgmt.purdue.edu., ERROR;(2) Department of Chemical Engineering, University of Illinois at Urbana-Champaign. e-mail: nikos@uiuc.edu., ERROR |
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Abstract: | We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function
over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions
under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex
extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest
convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving
convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited
by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that
are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms.
Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002
RID="*"
ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award
(DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S.
Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization |
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