Weighted Hypervolume Indicator

Authors: Johannes Bader, Dimo Brockhoff, and Eckart Zitzler

Implementation of hypervolume indicators for different weight distribution functions

Using the hypervolume of the dominated portion of the objective space as a measure for the quality of Pareto set approximations has received more and more attention in recent years.

So far, the hypervolume indicator is the only measure known in the literature on evolutionary multiobjective optimization that possesses the following two properties. On the one hand, it is sensitive to any type of improvements, i.e., whenever an approximation set A dominates another approximation set B, then the measure yields a strictly better quality value for the former than for the latter set. On the other hand, the hypervolume measure guarantees that any approximation set A that achieves the maximally possible quality value for a particular problem contains all Pareto-optimal objective vectors. With the recently proposed approach of a weighted hypervolume indicator, these properties are not removed by simultaneously be able to incorporate various user preferences.

According to [zbt2007a], three different hypervolume based indicators have been developed incorporating the following preferences:

1. Focus on extreme points,
2. Focus on the extremes of the second objective plus one additional extreme point for the first objective,
3. Focus on a given reference point.

All algorithms available on this website can be used for academic purposes without fee. If you publish research results achieved using the following (weighted) hypervolume indicator please remember to cite the corresponding EMO 2007 paper [zbt2007a]. The following copyright notice applies:

Copyright (c) 2006-2007 Swiss Federal Institute of Technology, Computer Engineering and Networks Laboratory. All rights reserved.

Permission to use, copy, modify, and distribute this software and its documentation for any purpose, without fee, and without written agreement is hereby granted, provided that the above copyright notice and the following two paragraphs appear in all copies of this software.

IN NO EVENT SHALL THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY, COMPUTER ENGINEERING AND NETWORKS LABORATORY BE LIABLE TO ANY PARTY FOR DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OF THIS SOFTWARE AND ITS DOCUMENTATION, EVEN IF THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY, COMPUTER ENGINEERING AND NETWORKS LABORATORY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY, COMPUTER ENGINEERING AND NETWORKS LABORATORY, SPECIFICALLY DISCLAIMS ANY WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE. THE SOFTWARE PROVIDED HEREUNDER IS ON AN "AS IS" BASIS, AND THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY, COMPUTER ENGINEERING AND NETWORKS LABORATORY HAS NO OBLIGATION TO PROVIDE MAINTENANCE, SUPPORT, UPDATES, ENHANCEMENTS, OR MODIFICATIONS.

The following implementation in C provides both the original hypervolume indicator algorithm of [zt1998b,ztlf2003a] as well as the three hypervolume indicators proposed in [zbt2007a].

Source code for the weighted hypervolume indicator as zip file (13 KB)

Source code for the weighted hypervolume indicator as tar file (46 KB)

[1 — zbt2007a]

E. Zitzler, D. Brockhoff, and L. Thiele. The Hypervolume Indicator Revisited: On the Design of Pareto-compliant Indicators Via Weighted Integration. In S. Obayashi et al., editors, Conference on Evolutionary Multi-Criterion Optimization (EMO 2007), volume 4403 of LNCS, pages 862–876, Berlin, 2007. Springer. (PDF) (bibtex) (online access) (suppl. material)

[1 — zt1998b]

E. Zitzler and L. Thiele. Multiobjective Optimization Using Evolutionary Algorithms - A Comparative Case Study. In Conference on Parallel Problem Solving from Nature (PPSN V), pages 292–301, Amsterdam, 1998. (PDF) (bibtex)

[1 — ztlf2003a]

E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. Grunert da Fonseca. Performance Assessment of Multiobjective Optimizers: An Analysis and Review. IEEE Transactions on Evolutionary Computation, 7(2):117–132, 2003. (PDF) (bibtex)