On the Construction of Pareto-Compliant Combined Indicators

The most relevant property that a quality indicator (QI) is expected to have is Pareto compliance, which means that every time an approximation set strictly dominates another in a Pareto sense, the indicator must reflect this. The hypervolume indicator and its variants are the only unary QIs known to be Pareto-compliant but there are many commonly used weakly Pareto-compliant indicators such as R2, IGD+ and ε+. Currently, an open research area is related to finding new Pareto-compliant indicators whose preferences are different to those of the hypervolume indicator. In this paper, we propose a theoretical basis to combine existing weakly Pareto-compliant indicators with at least one being Pareto-compliant, such that the resulting combined indicator is Pareto-compliant as well. Most importantly, we show that the combination of Paretocompliant QIs with weakly Pareto-compliant indicators leads to indicators that inherit properties of the weakly compliant indicators in terms of optimal point distributions. The consequences of these new combined indicators are threefold: 1) to increase the variety of available Pareto-compliant QIs by correcting weakly Pareto-compliant indicators, 2) to introduce a general framework for the combination of QIs, and 3) to generate new selection mechanisms for multi-objective evolutionary algorithms where it is possible to achieve/adjust desired distributions on the Pareto front.