Application of copulas to the estimation of Pareto fronts.

Summary It is common in optimization to start with a random draw in the space of variables to initialize a population or create a metamodel. In particular, in the multi-objective case, this leads to a set of non-dominated p oints that only inform p eu about the true Pareto front. We propose to study this problem from the point of view of multivariate analysis, by introducing a probabilistic framework and in particular by using copulas. Thus, expressions for the level lines are available in the objective space and consequently allow to obtain an estimate of the position of the Pareto front, when the level tends to zero. Explicit analytical expressions are available when Archimedean copulas are used. The corresponding estimation procedure is detailed and then applied on several examples.