Explicit convex and concave envelopes through polyhedral subdivisions

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....

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Bibliographic Details
Published inMathematical programming Vol. 138; no. 1-2; pp. 531 - 577
Main Authors Tawarmalani, Mohit, Richard, Jean-Philippe P., Xiong, Chuanhui
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer-Verlag 01.04.2013
Springer Nature B.V
Subjects
Algorithms
Calculus of Variations and Optimal Control; Optimization
Combinatorics
Construction
Envelopes
Exact solutions
Full Length Paper
Mathematical analysis
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematical programming
Mathematics
Mathematics and Statistics
Mathematics of Computing
Nonlinearity
Numerical Analysis
Optimization
Studies
Subdivisions
Theoretical
Variables
90C26
52A27
47N10
90C11
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Summary: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.
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ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-012-0581-4