On the estimation of Pareto fronts from the point of view of copula theory

Mickaël Binois (1, 2, 3, 4, 5, 6) , Didier Rullière (7) , Olivier Roustant (3, 2, 4, 5, 6)

1 Technocentre Renault [Guyancourt]
2 DEMO-ENSMSE - Département Décision en Entreprise : Modélisation, Optimisation
3 GdR MASCOT-NUM - Méthodes d'Analyse Stochastique des Codes et Traitements Numériques
4 LIMOS - Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes
5 Mines Saint-Étienne MSE - École des Mines de Saint-Étienne
6 FAYOL-ENSMSE - Institut Henri Fayol
7 SAF - Laboratoire de Sciences Actuarielle et Financière

Given a first set of observations from a design of experiments sampled randomly in the design space, the corresponding set of non-dominated points usually does not give a good approximation of the Pareto front. We propose here to study this problem from the point of view of multivariate analysis, introducing a probabilistic framework with the use of copulas. This approach enables the expression of level lines in the objective space, giving an estimation of the position of the Pareto front when the level tends to zero. In particular, when it is possible to use Archimedean copulas, analytical expressions for Pareto front estimators are available. Several case studies illustrate the interest of the approach, which can be used at the beginning of the optimization when sampling randomly in the design space.

Mots clés: Pareto front; Multi-objective optimization; copulas; Archimedean copulas

Domaines: Optimisation et contrôle [math.OC]; Probabilités [math.PR]; Applications [stat.AP]; Méthodes et statistiques

Mickaël Binois, Didier Rullière, Olivier Roustant. On the estimation of Pareto fronts from the point of view of copula theory. Information Sciences, 2015, 324, pp.270 - 285. ⟨10.1016/j.ins.2015.06.037⟩. ⟨hal-01097403v2⟩