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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)

Abstract

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.

Keywords

Pareto Front Multiobjective Optimization Convergence Speed Multiobjective Optimization Problem Nondominated Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  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)zbMATHGoogle 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)CrossRefzbMATHGoogle 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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