Pareto Cone ε-Dominance: Improving Convergence and Diversity in Multiobjective Evolutionary Algorithms

  • Lucas S. Batista
  • Felipe Campelo
  • Frederico G. Guimarães
  • Jaime A. Ramírez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6576)


Relaxed forms of Pareto dominance have been shown to be the most effective way in which evolutionary algorithms can progress towards the Pareto-optimal front with a widely spread distribution of solutions. A popular concept is the ε-dominance technique, which has been employed as an archive update strategy in some multiobjective evolutionary algorithms. In spite of the great usefulness of the ε-dominance concept, there are still difficulties in computing an appropriate value of ε that provides the desirable number of nondominated points. Additionally, several viable solutions may be lost depending on the hypergrid adopted, impacting the convergence and the diversity of the estimate set. We propose the concept of cone ε-dominance, which is a variant of the ε-dominance, to overcome these limitations. Cone ε-dominance maintains the good convergence properties of ε-dominance, provides a better control over the resolution of the estimated Pareto front, and also performs a better spread of solutions along the front. Experimental validation of the proposed cone ε-dominance shows a significant improvement in the diversity of solutions over both the regular Pareto-dominance and the ε-dominance.


Evolutionary multiobjective optimization evolutionary algorithms ε-dominance Pareto front 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the Strengh Pareto Evolutionary Algorithm. Tech. report 103, Computer Engineering and Networks Laboratory (2001)Google Scholar
  2. 2.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A Fast and Elitist Multiobjective Genetic Algorithm: NSGA-II. IEEE Trans. Evol. Comp. 6(2), 182–197 (2002)CrossRefGoogle Scholar
  3. 3.
    Zitzler, E., Thiele, L.: Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach. IEEE Trans. Evol. Comp. 3(4), 257–271 (1999)CrossRefGoogle Scholar
  4. 4.
    Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining Convergence and Diversity in Evolutionary Multi-Objective Optimization. Evolutionary Computation 10(3), 263–282 (2002)CrossRefGoogle Scholar
  5. 5.
    Hernández-Díaz, A.G., et al.: Pareto-Adaptive ε-Dominance. Evolutionary Computation 15(4), 493–517 (2007)CrossRefGoogle Scholar
  6. 6.
    Miettinen, K.M.: Nonlinear Multiobjective Optimization. International Series in Operations Research & Management Science. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  7. 7.
    Deb, K., Mohan, M., Mishra, S.: Towards a Quick Computation of Well-Spread Pareto-Optimal Solutions. In: Fonseca, C.M., et al. (eds.) EMO 2003. LNCS, vol. 2632, pp. 222–236. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Deb, K., Mohan, M., Mishra, S.: Evaluating the ε-Dominance Based Multi-Objective Evolutionary Algorithm for a Quick Computation of Pareto-Optimal Solutions. Evolutionary Computation 13(2), 501–525 (2005)CrossRefGoogle Scholar
  9. 9.
    Kanpur Genetic Algorithms Laboratory (KanGAL),
  10. 10.
    Deb, K.: Multi-Objective Genetic Algorithms: Problem Difficulties and Construction of Test Problems. Evolutionary Computation 7(3), 205–230 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Poloni, C.: Hybrid GA for Multiobjective Aerodynamic Shape Optimization. In: Winter, G., et al. (eds.) Genetic Algorithms in Engineering and Computer Science, pp. 397–416. Wiley, Chichester (1995)Google Scholar
  12. 12.
    Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation 8(2), 173–195 (2000)CrossRefGoogle Scholar
  13. 13.
    Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable Test Problems for Evolutionary Multiobjective Optimization. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 105–145. Springer, Heidelberg (2005)Google Scholar
  14. 14.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance Assessment of Multiobjective Optimizer: An Analysis and Review. IEEE Trans. Evol. Comp. 7(2), 117–132 (2003)CrossRefGoogle Scholar
  15. 15.
    Montgomery, D.C., Runger, G.C.: Applied Statistics and Probability for Engineers, 4th edn. Wiley, Chichester (2006)zbMATHGoogle Scholar
  16. 16.
    Hodges, J.L., Lehmann, E.L.: Estimation of location based on ranks. Annals of Mathematical Statistics 34, 598–611 (1963)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Lucas S. Batista
    • 1
  • Felipe Campelo
    • 1
  • Frederico G. Guimarães
    • 1
  • Jaime A. Ramírez
    • 1
  1. 1.Departamento de Engenharia ElétricaUniversidade Federal de Minas GeraisBelo HorizonteBrazil

Personalised recommendations