Choquet representability of submodular functions

Choquet representability of submodular functions Let $$\Omega $$ Ω be an arbitrary set, equipped with an algebra $${\mathcal {A}}\subseteq 2^{\Omega }$$ A ⊆ 2 Ω and let $$f: B({\mathcal {A}}) \rightarrow {\mathbb {R}}$$ f : B ( A ) → R be a functional defined on the set $$B({\mathcal {A}}) $$ B ( A ) of bounded measurable functions $$x:\Omega \rightarrow {\mathbb {R}}$$ x : Ω → R . We provide necessary and sufficient conditions for a submodular functional f to be representable as a Choquet integral. From standard properties of the Choquet integral the functional f should be positively homogeneous and constant additive. Our first result shows that these two properties, together with submodularity, characterize a subadditive Choquet integral, when $$\Omega $$ Ω is finite. In the general case, f is a subadditive Choquet integral if and only if it satisfies the three previous properties, together with sup-norm continuity. This result provides another characterization of subadditive Choquet integrals different from the seminal paper by Schmeidler (Proc Am Math Soc 97(2):255–261, 1986) that relies on comonotonic additivity. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematical Programming Springer Journals

Choquet representability of submodular functions

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2016 by Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Mathematics of Computing; Numerical Analysis; Combinatorics; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics
ISSN
0025-5610
eISSN
1436-4646
D.O.I.
10.1007/s10107-016-1074-7
Publisher site
See Article on Publisher Site

Abstract

Let $$\Omega $$ Ω be an arbitrary set, equipped with an algebra $${\mathcal {A}}\subseteq 2^{\Omega }$$ A ⊆ 2 Ω and let $$f: B({\mathcal {A}}) \rightarrow {\mathbb {R}}$$ f : B ( A ) → R be a functional defined on the set $$B({\mathcal {A}}) $$ B ( A ) of bounded measurable functions $$x:\Omega \rightarrow {\mathbb {R}}$$ x : Ω → R . We provide necessary and sufficient conditions for a submodular functional f to be representable as a Choquet integral. From standard properties of the Choquet integral the functional f should be positively homogeneous and constant additive. Our first result shows that these two properties, together with submodularity, characterize a subadditive Choquet integral, when $$\Omega $$ Ω is finite. In the general case, f is a subadditive Choquet integral if and only if it satisfies the three previous properties, together with sup-norm continuity. This result provides another characterization of subadditive Choquet integrals different from the seminal paper by Schmeidler (Proc Am Math Soc 97(2):255–261, 1986) that relies on comonotonic additivity.

Journal

Mathematical ProgrammingSpringer Journals

Published: Nov 8, 2016

References

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