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

• A probabilistic framework is applied to Pareto front estimation.
• A link between Pareto front and level lines of a multivariate distribution is given.
• Parametric approximations of Pareto front are provided for Archimedean copulas.
• Consequences and limits of the Archimedean hypothesis are discussed.
• The full estimation methodology is analyzed on several examples.

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.