Choquet integral optimisation with constraints and the buoyancy property for fuzzy measures

This work concerns solving optimisation problems where the objective function is expressed as a Choquet integral. This objective generalises a linear objective function (with positive weights) and allows for interaction to be modeled between coalitions of the deci-sion variables. We leverage results from optimising the ordered weighted averaging (OWA) operator and propose efficient solution approaches for the asymmetric objectives both for the simplest case of a single constraint and then for multiple comonotone constraints. To solve problems with a large number of variables, we rely on the so-called antibuoyancy property, previously applied to OWA weights, and which we extend to general fuzzy mea-sures. This characterisation not only facilitates a restriction of the domain on which the solution lies but also allows us to relate the Choquet integral's behavior in such cases to the Pigou-Dalton progressive transfers principle. We characterise the Choquet integrals consistent with the Pigou-Dalton principle. Theoretical results are supported by numerical experiments, which illustrate significant gains in performance. Our results offer opportuni-ties for scalability to a much higher number of variables. (c) 2021 Elsevier Inc. All rights reserved.

Automated summary

This work focuses on solving optimization problems using the Choquet integral as the objective function, which allows for interaction between coalitions of decision variables. Efficient solution approaches are proposed for problems with a large number of variables, leveraging the antibuoyancy property and extending it to general fuzzy measures. Theoretical results are supported by numerical experiments, showing significant performance gains and scalability to a higher number of variables.