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Explicit Convex and Concave Envelopes through Polyhedral Subdivisions

Mohit Tawarmalani(mtawarma***at***purdue.edu)
Jean-Philippe Richard(richard***at***ise.ufl.edu)
Chuanhui Xiong(cxiong***at***purdue.edu)

Abstract: In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.

Keywords: convex envelopes, supermodularity, disjunctive functions

Category 1: Global Optimization (Theory )

Category 2: Integer Programming ((Mixed) Integer Nonlinear Programming )

Category 3: Combinatorial Optimization (Polyhedra )

Citation: Krannert Working Paper

Download: [PDF]

Entry Submitted: 06/01/2010
Entry Accepted: 06/01/2010
Entry Last Modified: 06/01/2010

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