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Discrete representation of the non-dominated set for multi-objective optimization problems using kernels

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Date
2017
Dewey
Recherche opérationnelle
Sujet
Multiple objective programming; Pareto set; Non-dominated points; Discrete representation; Exact and approximation algorithms; kernel
JEL code
C.C4.C44
Journal issue
European Journal of Operational Research
Volume
260
Number
3
Publication date
2017
Article pages
814-827
DOI
http://dx.doi.org/10.1016/j.ejor.2016.11.020
URI
https://basepub.dauphine.fr/handle/123456789/16037
Collections
  • LAMSADE : Publications
Metadata
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Author
Bazgan, Cristina
989 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Jamain, Florian
989 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Vanderpooten, Daniel
989 Laboratoire d'analyse et modélisation de systèmes pour l'aide à la décision [LAMSADE]
Type
Article accepté pour publication ou publié
Abstract (EN)
In this paper, we are interested in producing discrete and tractable representations of the set of non-dominated points for multi-objective optimization problems, both in the continuous and discrete cases. These representations must satisfy some conditions of coverage, i.e. providing a good approximation of the non-dominated set, spacing, i.e. without redundancies, and cardinality, i.e. with the smallest possible number of points. This leads us to introduce the new concept of (ε, ε′)-kernels, or ε-kernels when ɛ′=ɛ is possible, which correspond to ε-Pareto sets satisfying an additional condition of ε′-stability. Among these, the kernels of small, or possibly optimal, cardinality are claimed to be good representations of the non-dominated set. We first establish some general properties on ε-kernels. Then, for the bi-objective case, we propose some generic algorithms computing in polynomial time either an ε-kernel of small size or, for a fixed size k , an ε-kernel with a nearly optimal approximation ratio 1+ɛ. For more than two objectives, we show that ε-kernels do not necessarily exist but that (ε, ε′)-kernels always exist. Nevertheless, we show that the size of a smallest (ε, ε′)-kernel can be very far from the size of a smallest ε-Pareto set.

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