A Study of Convergence Speed in Multi-objective Metaheuristics

  • Antonio Jesús Nebro
  • Juan José Durillo
  • Carlos A. Coello Coello
  • Francisco Luna
  • Enrique Alba
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5199)


An open issue in multi-objective optimization is designing metaheuristics that reach the Pareto front using a low number of function evaluations. In this paper, we adopt a benchmark composed of three well-known problem families (ZDT, DTLZ, and WFG) and analyze the behavior of six state-of-the-art multi-objective metaheuristics, namely, NSGA-II, SPEA2, PAES, OMOPSO, AbYSS, and MOCell, according to their convergence speed, i.e., the number of evaluations required to obtain an accurate Pareto front. By using the hypervolume as a quality indicator, we measure the algorithms converging faster, as well as their hit rate over 100 independent runs. Our study reveals that modern multi-objective metaheuristics such as MOCell, OMOPSO, and AbYSS provide the best overall performance, while NSGA-II and MOCell achieve the best hit rates.


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  1. 1.
    Blum, C., Roli, A.: Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Computing Surveys 35(3), 268–308 (2003)CrossRefGoogle Scholar
  2. 2.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002)CrossRefGoogle Scholar
  3. 3.
    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. Theoretical Advances and Applications, pp. 105–145. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  4. 4.
    Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms. John Wiley & Sons, Chichester (2001)MATHGoogle Scholar
  5. 5.
    Durillo, J.J., Nebro, A.J., Luna, F., Dorronsoro, B., Alba, E.: jMetal: a java framework for developing multi-objective optimization metaheuristics. Technical Report ITI-2006-10, Dpto. de Lenguajes y Ciencias de la Computación, University of Málaga (2006)Google Scholar
  6. 6.
    Eskandari, H., Geiger, C.D., Lamont, G.B.: FastPGA: A dynamic population sizing approach for solving expensive multiobjective optimization problems. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 141–155. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Glover, F., Kochenberger, G.A.: Handbook of Metaheuristics. Kluwer Academic Publishers, Dordrecht (2003)CrossRefMATHGoogle Scholar
  8. 8.
    Hernández-Díaz, A.G., Santana-Quintero, L.V., Coello Coello, C., Caballero, R., Molina, J.: A New Proposal for Multi-Objective Optimization using Differential Evolution and Rough Sets Theory. In: Keijzer, M., et al. (eds.) Genetic and Evolutionary Computation Conference (GECCO 2006), pp. 675–682 (2006)Google Scholar
  9. 9.
    Huband, S., Hingston, P., Barone, L., While, L.: A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit. IEEE Transactions on Evolutionary Computation 10(5), 477–506 (2006)CrossRefGoogle Scholar
  10. 10.
    Knowles, J.: ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Transactions on Evolutionary Computation 10(1), 50–66 (2006)CrossRefGoogle Scholar
  11. 11.
    Knowles, J., Thiele, L., Zitzler, E.: A tutorial on the performance assessment of stochastic multiobjective optimizers. TIK Report 214, Computer Engineering and Networks Laboratory (TIK), ETH Zurich (2006)Google Scholar
  12. 12.
    Knowles, J.D., Corne, D.W.: The Pareto Archived Evolution Strategy: A New Baseline Algorithm for Multiobjective Optimisation. In: Congress on Evolutionary Computation, pp. 98–105 (1999)Google Scholar
  13. 13.
    Nebro, A.J., Durillo, J.J., Luna, F., Dorronsoro, B., Alba, E.: A cellular genetic algorithm for multiobjective optimization. In: Nature Inspired Cooperative Strategies for Optimization (NICSO 2006), pp. 25–36 (2006)Google Scholar
  14. 14.
    Nebro, A.J., Durillo, J.J., Luna, F., Dorronsoro, B., Alba, E.: Design issues in a multiobjective cellular genetic algorithm. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 126–140. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  15. 15.
    Nebro, A.J., Luna, F., Alba, E., Dorronsoro, B., Durillo, J.J., Beham, A.: AbYSS: Adapting scatter search to multiobjective optimization. IEEE Transactions on Evolutionary Computation (to appear, 2008)Google Scholar
  16. 16.
    Reyes Sierra, M., Coello Coello, C.A.: Improving PSO-Based Multi-objective Optimization Using Crowding, Mutation and ε-Dominance. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 505–519. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Santana-Quintero, L.V., Ramírez-Santiago, N., Coello Coello, C.A., Molina Luque, J., García Hernández-Díaz, A.: A New Proposal for Multiobjective Optimization Using Particle Swarm Optimization and Rough Sets Theory. In: Runarsson, T.P., Beyer, H.-G., Burke, E.K., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) PPSN IX 2006. LNCS, vol. 4193, pp. 483–492. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Toscano-Pulido, G., Coello Coello, C.A., Santana-Quintero, L.V.: EMOPSO: A Multi-Objective Particle Swarm Optimizer with Emphasis on Efficiency. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 272–285. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  19. 19.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation 8(2), 173–195 (2000)CrossRefGoogle Scholar
  20. 20.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the Strength Pareto Evolutionary Algorithm. In: EUROGEN 2001, pp. 95–100 (2002)Google Scholar
  21. 21.
    Zitzler, E., Thiele, L.: Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach. IEEE Transactions on Evolutionary Computation 3(4), 257–271 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Antonio Jesús Nebro
    • 1
  • Juan José Durillo
    • 1
  • Carlos A. Coello Coello
    • 2
  • Francisco Luna
    • 1
  • Enrique Alba
    • 1
  1. 1.Department of Computer ScienceUniversity of MálagaSpain
  2. 2.Department of Computer ScienceCINVESTAV-IPNMexico

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