Hypervolume-based expected improvement: Monotonicity properties and exact computation

The expected improvement (EI) is a well established criterion in Bayesian global optimization (BGO) and metamodel assisted evolutionary computation, both applied in optimization with costly function evaluations. Recently, it has been adopted in different ways to multiobjective optimization. A promising approach to formulate the expected improvement in this context, is to base it on the hypervolume indicator. Given the Bayesian model of the optimization landscape, the EI in hypervolume computes the expected gain in attained hypervolume for a given input point. Although a formulation of this expected improvement is relatively straightforward, its computation and mathematical properties are still to be investigated. This paper will outline and derive an algorithm for the exact computation of the proposed hypervolume-based EI. Moreover, this paper establishes monotonicity properties of the expected improvement. In particular the effect of the predictive distribution's variance on the hypervolume-based EI and elementary properties of the EI landscape are studied. The monotonicity properties will reveal regions where Pareto front approximations can be improved as well as underexplored regions that are favored by the hypervolume based expected improvement. A first numerical example is included that illustrates the behavior of the hypervolume-based EI in the multiobjective BGO framework.