On the convex hull of K-additive 0-1 capacities and its application to model identification in decision making

Michel Grabisch and Christophe Labreuche ()
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Christophe Labreuche: Thales Research & Technology - Palaiseau, SINCLAIR AI Lab - Palaiseau

Documents de travail du Centre d'Economie de la Sorbonne from Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne

Abstract: The choquet integral w.r.t. a capacity is a versatile tool commonly used in decision making. Its practical identification requires, however, to solve an optimization problem with exponentially many variables and constraints. The introduction of k-additive capacities, through the use of the Möbius transform, permits to reduce the number of variables to a polynomial size, but leaves the number of constraints exponential. When k = 2, the use of vertices of the set of 2-additive capacities permits to solve the problem as the number of vertices is polynomial. When k > 2, this solution is no more applicable as the set of vertices of k-additive capacities is not known. We propose in this paper to use instead the set of vertices which are 0-1 valued. We show that the loss of generality is small, and that the number of such vertices is polynomial. Also, we study the geometric properties of the convex hull of 0-1 valued k-additive capacities

Keywords: capacity; k-additive capacity; Choquet integral; vertices; facets (search for similar items in EconPapers)
JEL-codes: C71 (search for similar items in EconPapers)
Pages: 28 pages
Date: 2022-01
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Published in Fuzzy Sets and Systems, Elsevier, 451, 2022, pp. 228-252

Downloads: (external link)
http://mse.univ-paris1.fr/pub/mse/CES2022/22003.pdf (application/pdf)
https://shs.hal.science/halshs-03561127
https://doi.org/10.1016/j.fss.2022.03.018

Related works:
Working Paper: On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making (2022)
Working Paper: On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making (2022)
Working Paper: On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making (2022)
Working Paper: On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making (2022)
Working Paper: On the convex hull of k-additive 0-1 capacities and its application to model identification in decision making (2022)
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