Hypervolume-based Multi-Objective Expected Improvement for Three-Objective Functions

This paper proposes a methodology for the direct quadrature of hypervolume-based expected improvement when objective spaces have more than two dimensions. New terms with respect to the hypervolume-based expected improvement for the-two objective case are explained and derived. When adaptively sampling computationally intensive multi-objective domains, the proposed methodology represents an alternative to the inaccurate and resource-consuming current state-of-the-art method, Monte Carlo integration, for the quadrature calculation of the hypervolume-based expected improvement. The methodology is first compared with the current state-of-the-art one on typical multi-objective canonical problems. This comparison allows to determine in which conditions each of the methodologies is more competitive while adaptively sampling multi-objective domains of computationally intensive functions. Next, a practical design space of an engineering system is adaptively sampled using the proposed methodology for the quadrature.