Abstract
The hypervolume indicator (HV) has been subject of a lot of research in the last few years, mainly because its maximization yields near-optimal approximations of the Pareto optimal front of a multi-objective optimization problem. This feature has been exploited by several evolutionary optimizers, in spite of the considerable growth in computational cost that it is involved in the computation of HV as we increase the number of objectives. Some years ago, the Walking Fish Group (WFG) implemented a new version of the incremental hypervolume algorithm, named IWFG 1.01. This implementation is the fastest reported to date for determining the solution that contributes the least to the HV of a non-dominated set. Nevertheless, this new version has gone mostly unnoticed by the research community. We believe that this is due to an error in the source code provided by the authors of this algorithm, which appears when coupling it to a multi-objective evolutionary algorithm. In this paper, we describe this error, and we propose a solution to fix it. Moreover, we illustrate the significant gains in performance produced by IWFG 1.01 in many-objective optimization problems (i.e., problems having three or more objectives), when integrated into the S-Metric Selection Evolutionary Multi-Objective Algorithm (SMS-EMOA).
The third author acknowledges support from CONACyT project no. 2016-01-1920 (Investigación en Fronteras de la Ciencia 2016).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
For instance, \(\text {head}([a,b,c,d]):= a\) and \(\text {tail}([a,b,c,d]):= [b,c,d]\).
References
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181(3), 1653–1669 (2007)
Bradstreet, L., While, L., Barone, L.: A new way of calculating exact exclusive hypervolumes. Technical report, UWA-CSSE-09-002, The University of Western Australia, School of Computer Science and Software Engineering (2009)
Bradstreet, L., While, L., Barone, L.: A fast incremental hypervolume algorithm. IEEE Trans. Evol. Comput. 12(6), 714–723 (2008)
Coello Coello, C.A., Lamont, G.B., Van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-Objective Problems, 2nd edn. Springer, New York (2007). https://doi.org/10.1007/978-0-387-36797-2. ISBN 978-0-387-33254-3
Cox, W., While, L.: Improving the IWFG algorithm for calculating incremental hypervolume. In: 2016 IEEE Congress on Evolutionary Computation (CEC), pp. 3969–3976, July 2016
Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Trans. Evol. Comput. 18(4), 577–601 (2014)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable test problems for evolutionary multiobjective optimization. In: Abraham, A., Jain, L., Goldberg, R. (eds.) Evolutionary Multiobjective Optimization, pp. 105–145. Advanced Information and Knowledge Processing, Springer (2005). https://doi.org/10.1007/1-84628-137-7_6
Falcón-Cardona, J.G., Coello, C.A.C.: Indicator-based multi-objective evolutionary algorithms: a comprehensive survey. ACM Comput. Surv. 53(2) (2020). https://doi.org/10.1145/3376916
Fleischer, M.: The measure of pareto optima applications to multi-objective metaheuristics. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Thiele, L., Deb, K. (eds.) EMO 2003. LNCS, vol. 2632, pp. 519–533. Springer, Heidelberg (2003). https://doi.org/10.1007/3-540-36970-8_37
Hernández Gómez, R.: Parallel hyper-heuristics for multi-objective optimization. Ph.D. thesis, CINVESTAV-IPN, Mexico City, Mexico, March 2018
Tan, Y., Jiao, Y., Li, H., Wang, X.: MOEA/D plus uniform design: a new version of MOEA/D for optimization problems with many objectives. Comput. Oper. Res. 40(6), 1648–1660 (2013)
While, L., Bradstreet, L., Barone, L.: A fast way of calculating exact hypervolumes. IEEE Trans. Evol. Comput. 16(1), 86–95 (2012)
While, L., Bradstreet, L.: Applying the WFG algorithm to calculate incremental hypervolumes. In: 2012 IEEE Congress on Evolutionary Computation (CEC 2012), Brisbane, Australia, 10–15 June 2012, pp. 489–496. IEEE Press (2012)
Zitzler, E.: Evolutionary algorithms for multiobjective optimization: methods and applications. Ph.D. thesis, Swiss Federal Institute of Technology (ETH), Zurich, Switzerland, November 1999
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Hernández Gómez, R., Falcón-Cardona, J.G., Coello, C.A.C. (2022). Considerations in the Incremental Hypervolume Algorithm of the WFG. In: Pichardo Lagunas, O., Martínez-Miranda, J., Martínez Seis, B. (eds) Advances in Computational Intelligence. MICAI 2022. Lecture Notes in Computer Science(), vol 13612. Springer, Cham. https://doi.org/10.1007/978-3-031-19493-1_32
Download citation
DOI: https://doi.org/10.1007/978-3-031-19493-1_32
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-19492-4
Online ISBN: 978-3-031-19493-1
eBook Packages: Computer ScienceComputer Science (R0)