On the Construction of Pareto-Compliant Combined Indicators

Author:

Falcón-Cardona J. G.1,Emmerich M. T. M.2,Coello C. A. Coello3

Affiliation:

1. Computer Science Department, CINVESTAV-IPN, Mexico City, 07360, Mexico jfalcon@computacion.cs.cinvestav.mx

2. LIACS, Leiden University, Leiden, 2333, The Netherlands m.t.m.emmerich@liacs.leidenuniv.nl

3. Computer Science Department, CINVESTAV-IPN, Mexico City, 07360, Mexico Basque Center for Applied Mathematics (BCAM) & Ikerbasque, Spain ccoello@cs.cinvestav.mx

Abstract

Abstract 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.

Publisher

MIT Press

Subject

Computational Mathematics

Reference54 articles.

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3. An empirical assessment of the properties of inverted generational distance on multi- and many-objective optimization;Bezerra;Proceedings of the 9th International Conference on Evolutionary Multi-Criterion Optimization,2017

4. Approximating the least hypervolume contributor: NP-hard in general, but fast in practice;Bringmann;Fifth International Conference on Evolutionary Multi-Criterion Optimization,2009

5. Approximating the volume of unions and intersections of high-dimensional geometric objects;Bringmann;Computational Geometry---Theory and Applications,2010

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