# Quasi-Lovász Extensions and Their Symmetric Counterparts

2 ORPAILLEUR - Knowledge representation, reasonning
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
Abstract : We introduce the concept of quasi-Lov\'asz extension as being a mapping $f\colon I^n\to\mathbb{R}$ defined over a nonempty real interval $I$ containing the origin, and which can be factorized as $f(x_1,\ldots,x_n)=L(\varphi(x_1),\ldots,\varphi(x_n))$, where $L$ is the Lov\'asz extension of a pseudo-Boolean function $\psi\colon\{0,1\}^n\to\mathbb{R}$ (i.e., the function $L\colon\mathbb{R}^n\to\mathbb{R}$ whose restriction to each simplex of the standard triangulation of $[0,1]^n$ is the unique affine function which agrees with $\psi$ at the vertices of this simplex) and $\varphi\colon I\to\mathbb{R}$ is a nondecreasing function vanishing at the origin. These functions appear naturally within the scope of decision making under uncertainty since they subsume overall preference functionals associated with discrete Choquet integrals whose variables are transformed by a given utility function. To axiomatize the class of quasi-Lov\'asz extensions, we propose generalizations of properties used to characterize the Lov\'asz extensions, including a comonotonic version of modularity and a natural relaxation of homogeneity. A variant of the latter property enables us to axiomatize also the class of symmetric quasi-Lov\'asz extensions, which are compositions of symmetric Lov\'asz extensions with $1$-place nondecreasing odd functions.
Document type :
Conference papers
Domain :

https://hal.archives-ouvertes.fr/hal-01093650
Contributor : Miguel Couceiro <>
Submitted on : Saturday, February 18, 2017 - 10:39:49 AM
Last modification on : Tuesday, December 18, 2018 - 4:38:02 PM
Long-term archiving on : Friday, May 19, 2017 - 12:10:56 PM

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• HAL Id : hal-01093650, version 1

### Citation

Miguel Couceiro, Jean-Luc Marichal. Quasi-Lovász Extensions and Their Symmetric Counterparts. 14th International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, IPMU 2012, Jul 2012, Catania, Italy. ⟨hal-01093650⟩

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