Pareto-, Aggregation-, and Indicator-Based Methods in Many-Objective Optimization

  • Tobias Wagner
  • Nicola Beume
  • Boris Naujoks
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4403)


Research within the area of Evolutionary Multi-objective Optimization (EMO) focused on two- and three-dimensional objective functions, so far. Most algorithms have been developed for and tested on this limited application area. To broaden the insight in the behavior of EMO algorithms (EMOA) in higher dimensional objective spaces, a comprehensive benchmarking is presented, featuring several state-of-the-art EMOA, as well as an aggregative approach and a restart strategy on established scalable test problems with three to six objectives. It is demonstrated why the performance of well-established EMOA (NSGA-II, SPEA2) rapidly degradates with increasing dimension. Newer EMOA like ε-MOEA, MSOPS, IBEA and SMS-EMOA cope very well with high-dimensional objective spaces. Their specific advantages and drawbacks are illustrated, thus giving valuable hints for practitioners which EMOA to choose depending on the optimization scenario. Additionally, a new method for the generation of weight vectors usable in aggregation methods is presented.


Weight Vector Pareto Front Objective Space Target Vector Extremal 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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  1. 1.
    Farina, M., Amato, P.: On the optimal solution definition for many-criteria optimization problems. In: Keller, J., Nasraoui, O. (eds.) Proc. of the NAFIPS-FLINT Int’l Conf. 2002, pp. 233–238. IEEE Computer Society Press, Piscataway (2002)Google Scholar
  2. 2.
    Deb, K.: Multi-Objective Optimization using Evolutionary Algorithms. Wiley, Chichester (2001)zbMATHGoogle Scholar
  3. 3.
    Coello Coello, C.A., Van Veldhuizen, D.A., Lamont, G.B.: Evolutionary Algorithms for Solving Multi-Objective Problems. Kluwer, New York (2002)zbMATHGoogle Scholar
  4. 4.
    Purshouse, R.C., Fleming, P.J.: Evolutionary Multi-Objective Optimisation: An Exploratory Analysis. In: Proc. of the 2003 Congress on Evolutionary Computation (CEC’2003), vol. 3, Canberra, Australia, pp. 2066–2073. IEEE Press, Los Alamitos (2003)CrossRefGoogle Scholar
  5. 5.
    Hughes, E.J.: Evolutionary Many-Objective Optimisation: Many Once or One Many? In: Evolutionary Computation ’Congress (CEC’05), Edinburgh, UK, vol. 1, pp. 222–227. IEEE Press, Piscataway (2005)CrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Bleuler, S., Laumanns, M., Thiele, L., Zitzler, E.: PISA — a platform and programming language independent interface for search algorithms. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 494–508. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  8. 8.
    Deb, K., Mohan, M., Mishra, S.: A Fast Multi-objective Evolutionary Algorithm for Finding Well-Spread Pareto-Optimal Solutions. KanGAL report 2003002, Indian Institute of Technology, Kanpur, India (2003)Google Scholar
  9. 9.
    Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable Multi-objective Optimization Test Problems. In: Proc. of the 2002 Congress on Evolutionary Computation (CEC 2002), vol. 1, pp. 825–830. IEEE Press, Piscataway (2002)Google Scholar
  10. 10.
    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.) Parallel Problem Solving from Nature - PPSN V. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  11. 11.
    Naujoks, B., Beume, N., Emmerich, M.: Multi-objective optimisation using S-metric selection: Application to three-dimensional solution spaces. In: Evolutionary Computation Congress (CEC’05), Edinburgh, UK, pp. 1282–1289. IEEE Press, Piscataway (2005)CrossRefGoogle Scholar
  12. 12.
    Jensen, M.T.: Reducing the run-time complexity of multiobjective EAs: The NSGA-II and other algorithms. IEEE Transactions On Evolutionary Computation 7(5), 503–515 (2003)CrossRefGoogle Scholar
  13. 13.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the Strength Pareto Evolutionary Algorithm. Technical Report 103, Computer Engineering and Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH) Zürich, Switzerland (2001)Google Scholar
  14. 14.
    Silverman, B.W.: Density estimation for statistics and data analysis. Chapman and Hall, London (1986)zbMATHGoogle Scholar
  15. 15.
    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
  16. 16.
    Rudolph, G., Agapie, A.: Convergence properties of some multi-objective evolutionary algorithms. In: Zalzala, A., Eberhart, R. (eds.) Congress on Evolutionary Computation (CEC2000), vol. 2, pp. 1010–1016. IEEE Press, Piscataway (2000)Google Scholar
  17. 17.
    Hughes, E.J.: Multiple Single Objective Pareto Sampling. In: Congress on Evolutionary Computation (CEC’03), IEEE Press, Piscataway (2003)Google Scholar
  18. 18.
    Ostermeier, A., Gawelczyk, A., Hansen, N.: Step-size adaptation based on non-local use of selection information. In: Davidor, Y., Männer, R., Schwefel, H.-P. (eds.) Parallel Problem Solving from Nature - PPSN III. LNCS, vol. 866, pp. 189–198. Springer, Heidelberg (1994)Google Scholar
  19. 19.
    Hansen, N., Ostermeier, A.: Completely Derandomized Self-Adaptation in Evolution Strategies. IEEE Computational Intelligence Magazine 9(2), 159–195 (2001)Google Scholar
  20. 20.
    Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) Parallel Problem Solving from Nature - PPSN VIII. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004)Google Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    Beume, N., Rudolph, G.: Faster S-Metric Calculation by Considering Dominated Hypervolume as Klee’s Measure Problem. In: International Conference on Computational Intelligence (CI (2006) (in print, 2006)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Tobias Wagner
    • 1
  • Nicola Beume
    • 2
  • Boris Naujoks
    • 2
  1. 1.Institut für Spanende Fertigung (ISF) 
  2. 2.Chair of Algorithm Engineering, University of Dortmund, 44221 DortmundGermany

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