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 from those of the hypervolume indicator. In this article, 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 Pareto-compliant 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 multiobjective evolutionary algorithms where it is possible to achieve/adjust desired distributions on the Pareto front.

Search Dynamics on Multimodal Multiobjective Problems.

We continue recent work on the definition of multimodality in multiobjective optimization (MO) and the introduction of a test bed for multimodal MO problems. This goes beyond well-known diversity maintenance approaches but instead focuses on the landscape topology induced by the objective functions. More general multimodal MO problems are considered by allowing ellipsoid contours for single-objective subproblems. An experimental analysis compares two MO algorithms, one that explicitly relies on hypervolume gradient approximation, and one that is based on local search, both on a selection of generated example problems. We do not focus on performance but on the interaction induced by the problems and algorithms, which can be described by means of specific characteristics explicitly designed for the multimodal MO setting. Furthermore, we widen the scope of our analysis by additionally applying visualization techniques in the decision space. This strengthens and extends the foundations for Exploratory Landscape Analysis (ELA) in MO.