The computation of the expected improvement in dominated hypervolume of Pareto front approximations

Abstract

The hypervolume measure is used frequently in the design and performance assessment of multiobjective optimization algorithms. Especially in the context of metamodel (or surrogate) assisted optimization [1, 2] it is interesting to look at the following problem. Given an approximation set for the Pareto front and a new candidate solution x that has not yet been evaluated precisely but for which a prediction with uncertainty measure in the form of an independent multivariate Gaussian distribution with mean vector ~ and standard deviation vector ~ exists (Figure 1): What is the expected improvement of the hypervolume when x is being added to the population? For multiple objectives up til now only a Monte Carlo method has been reported. This contribution provides a direct computation procedure for the integral expression. This will be useful to enhance both accuracy and speed of computation for this important measure.