On Reference Point Free Weighted Hypervolume Indicators based on Desirability Functions and their Probabilistic Interpretation

Abstract The so-called a posteriori approach to optimization with multiple conflicting objective functions is to compute or approximate a Pareto front of solutions. In case of continuous objective functions a finite approximation to this set can be computed. Indicator-based multiobjective optimization algorithms compute solution sets that are optimal with respect to some quality measure on sets, such as the commonly used hypervolume indicator (HI). The HI measures the size of the space that is dominated by a given set of solutions. It has many favorable monotonicity properties but it requires a reference point the choice of which is often done ad-hoc. In this study the concept of set monotonic functions for dominated subsets is introduced. Moreover, this work presents a reference point free hypervolume indicator that uses a density that is derived from the user's preferences expressed as desirability functions. This approach will bias the distribution of the approximation set towards a set that more densely samples highly desirable solutions of the objective space. We show that the Harrington type and the Derringer-Suich type of desirability functions yield definite integrals and that the Harrington type has also the favorable property to provide a set-monotonic function over the set of dominated subspaces. It is shown that for a product type of aggregation the weighted hypervolume indicator is mathematically equivalent with an approach that computes the standard hypervolume indicator after transformation of the axes. In addition a probabilistic interpretation of desirability functions is discussed and how a correlation parameter can be introduced in order to change the aggregation type. Finally, practical guidelines for using the discussed set indicator in multiobjective search, for instance when searching for interesting subsets from a database, are provided.