Abstract
Hypervolume (HV) and inverted generational distance (IGD) have been frequently used as performance indicators to evaluate the quality of solution sets obtained by evolutionary multiobjective optimization (EMO) algorithms. They have also been used in indicator-based EMO algorithms. In some studies on many-objective problems, only the IGD indicator was used due to a large computation load of HV calculation. However, the IGD indicator is not Pareto compliant. This means that a better solution set in terms of the Pareto dominance relation can be evaluated as being worse. Recently the IGD plus (IGD+) indicator has been proposed as a weakly Pareto compliant version of IGD. In this paper, we compare these three indicators from the viewpoint of optimal distributions of solutions. More specifically, we visually demonstrate similarities and differences among the three indicators by numerically calculating near-optimal distributions of solutions to optimize each indicator for some test problems. Our numerical analysis shows that IGD+ is more similar to HV than IGD whereas the formulations of IGD and IGD+ are almost the same.
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References
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Hypervolume-based multiobjective optimization: theoretical foundations and practical implications. Theoret. Comput. Sci. 425, 75–103 (2012)
Bader, J., Zitzler, E.: HypE: an algorithm for fast hypervolume-based many-objective optimization. Evol. Comput. 19, 45–76 (2011)
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181, 1653–1669 (2007)
Brockhoff, D.: Optimal μ-distributions for the hypervolume indicator for problems with linear bi-objective fronts: exact and exhaustive results. In: Deb, K., et al. (eds.) SEAL 2010. LNCS, vol. 6457, pp. 24–34. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-17298-4_2
Coello Coello, C.A., Reyes Sierra, M.: A study of the parallelization of a coevolutionary multi-objective evolutionary algorithm. In: Monroy, R., Arroyo-Figueroa, G., Sucar, L.E., Sossa, H. (eds.) MICAI 2004. LNCS, vol. 2972, pp. 688–697. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24694-7_71
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 182–197 (2002)
Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based non-dominated sorting approach, part I: solving problems with box constraints. IEEE Trans. Evol. Comput. 18, 577–601 (2014)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable multi-objective optimization test problems. In: Proceedings of IEEE CEC 2002, pp. 825–830 (2002)
Emmerich, M., Beume, N., Naujoks, B.: An EMO algorithm using the hypervolume measure as selection criterion. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 62–76. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-31880-4_5
Emmerich, M., Deutz, A., Beume, N.: Gradient-based/evolutionary relay hybrid for computing Pareto front approximations maximizing the S-metric. In: Bartz-Beielstein, T., et al. (eds.) HM 2007. LNCS, vol. 4771, pp. 140–156. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-75514-2_11
Huband, S., Hingston, P., Barone, L., While, L.: A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans. Evol. Comput. 10, 477–506 (2006)
Ishibuchi, H., Imada, R., Setoguchi, Y., Nojima, Y.: Hypervolume subset selection for triangular and inverted triangular Pareto fronts of three-objective problems. In: Proceedings of FOGA 2017, pp. 95–110 (2017)
Ishibuchi, H., Imada, R., Setoguchi, Y., Nojima, Y.: How to specify a reference point in hypervolume calculation for fair performance comparison. Evol. Comput. 26, 411–440 (2018)
Ishibuchi, H., Imada, R., Setoguchi, Y., Nojima, Y.: Reference point specification in inverted generational distance for triangular linear Pareto front. IEEE Trans. Evol. Comput. 22, 961–975 (2018)
Ishibuchi, H., Masuda, H., Tanigaki, Y., Nojima, Y.: Modified distance calculation in generational distance and inverted generational distance. In: Gaspar-Cunha, A., Henggeler Antunes, C., Coello, C.C. (eds.) EMO 2015. LNCS, vol. 9019, pp. 110–125. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15892-1_8
Ishibuchi, H., Setoguchi, Y., Masuda, H., Nojima, Y.: Performance of decomposition based many-objective algorithms strongly depends on Pareto front shapes. IEEE Trans. Evol. Comput. 21, 169–190 (2017)
Jain, H., Deb, K.: An improved adaptive approach for elitist nondominated sorting genetic algorithm for many-objective optimization. In: Purshouse, R.C., Fleming, P.J., Fonseca, C.M., Greco, S., Shaw, J. (eds.) EMO 2013. LNCS, vol. 7811, pp. 307–321. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-37140-0_25
Jiang, S.W., Ong, Y.-S., Zhang, J., Feng, L.: Consistencies and contradictions of performance metrics in multiobjective optimization. IEEE Trans. Cybern. 44, 2391–2404 (2014)
Jiang, S.W., Zhang, J., Ong, Y.-S., Zhang, A.N., Tan, P.S.: A simple and fast hypervolume indicator-based multiobjective evolutionary algorithm. IEEE Trans. Cybern. 45, 2202–2213 (2015)
Lopez, E.M., Coello Coello, C.A.: An improved version of a reference-based multi-objective evolutionary algorithm based on IGD+. In: Proceedings of GECCO 2018, pp. 713–720 (2018)
Ravber, M., Mernik, M., Crepinkek, M.: The impact of quality indicators on the rating of multi-objective evolutionary algorithms. Appl. Soft Comput. 55, 265–275 (2017)
Sierra, M.R., Coello Coello, C.A.: A new multi-objective particle swarm optimizer with improved selection and diversity mechanisms. Technical report. CINVESTAV-IPN (2004)
Sun, Y., Yen, G.G., Yi, Z.: IGD indicator-based evolutionary algorithm for many-objective optimization problems. IEEE Trans. Evol. Comput. (Early Access Paper: Online Available)
Tian, Y., Cheng, R., Zhang, X.Y., Cheng, F., Jin, Y.C.: An indicator-based multiobjective evolutionary algorithm with reference point adaptation for better versatility. IEEE Trans. Evol. Comput. 22, 609–622 (2018)
Wagner, T., Beume, N., Naujoks, B.: Pareto-, aggregation-, and indicator-based methods in many-objective optimization. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 742–756. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70928-2_56
Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11, 712–731 (2007)
Zitzler, E., Brockhoff, D., Thiele, L.: The hypervolume indicator revisited: on the design of Pareto-compliant indicators via weighted integration. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 862–876. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70928-2_64
Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms—a comparative case study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998). https://doi.org/10.1007/BFb0056872
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7, 117–132 (2003)
Acknowledgments
This work was supported by National Natural Science Foundation of China (Grant No. 61876075), the Program for Guangdong Introducing Innovative and Entrepreneurial Teams (Grant No. 2017ZT07X386), Shenzhen Peacock Plan (Grant No. KQTD2016112514355531), the Science and Technology Innovation Committee Foundation of Shenzhen (Grant No. ZDSYS201703031748284), and the Program for University Key Laboratory of Guangdong Province (Grant No. 2017KSYS008).
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Ishibuchi, H., Imada, R., Masuyama, N., Nojima, Y. (2019). Comparison of Hypervolume, IGD and IGD+ from the Viewpoint of Optimal Distributions of Solutions. In: Deb, K., et al. Evolutionary Multi-Criterion Optimization. EMO 2019. Lecture Notes in Computer Science(), vol 11411. Springer, Cham. https://doi.org/10.1007/978-3-030-12598-1_27
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