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 [France], SINCLAIR AI Lab - Saclay Industrial Lab for Artificial Intelligence Research - THALES [France] - EDF - EDF - [Total Energies. Anciennement : Total, TotalFina, TotalFinaElf] - TotalEnergies

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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 number of such vertices is polynomial, and we observe that the loss of generality is very small for n = 4, k = 3, and conjecture that this still holds for larger values of n. 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; capacity; k-additive capacity; Choquet integral; vertices; facets (search for similar items in EconPapers)
Date: 2022-12-28
Note: View the original document on HAL open archive server: https://shs.hal.science/halshs-03881431v1
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Published in Fuzzy Sets and Systems, 2022, 451, pp.228-252. ⟨10.1016/j.fss.2022.03.018⟩

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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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Persistent link: https://EconPapers.repec.org/RePEc:hal:pseptp:halshs-03881431

DOI: 10.1016/j.fss.2022.03.018

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