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On Sequential Online Archiving of Objective Vectors

  • Manuel López-Ibáñez
  • Joshua Knowles
  • Marco Laumanns
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6576)

Abstract

In this paper, we examine the problem of maintaining an approximation of the set of nondominated points visited during a multiobjective optimization, a problem commonly known as archiving. Most of the currently available archiving algorithms are reviewed, and what is known about their convergence and approximation properties is summarized. The main scenario considered is the restricted case where the archive must be updated online as points are generated one by one, and at most a fixed number of points are to be stored in the archive at any one time. In this scenario, the \(\vartriangleleft\)-monotonicity of an archiving algorithm is proposed as a weaker, but more practical, property than negative efficiency preservation. This paper shows that hypervolume-based archivers and a recently proposed multi-level grid archiver have this property. On the other hand, the archiving methods used by SPEA2 and NSGA-II do not, and they may \(\vartriangleleft\)-deteriorate with time. The \(\vartriangleleft\)-monotonicity property has meaning on any input sequence of points. We also classify archivers according to limit properties, i.e. convergence and approximation properties of the archiver in the limit of infinite (input) samples from a finite space with strictly positive generation probabilities for all points. This paper establishes a number of research questions, and provides the initial framework and analysis for answering them.

Keywords

approximation set archive convergence efficiency preserving epsilon-dominance hypervolume online algorithms 

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References

  1. 1.
    Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research 181(3), 1653–1669 (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    Borodin, A., El-Yaniv, R.: Online computation and competitive analysis. Cambridge University Press, New York (1998)zbMATHGoogle Scholar
  3. 3.
    Bringmann, K., Friedrich, T.: Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 6–20. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  4. 4.
    Bringmann, K., Friedrich, T.: Don’t be greedy when calculating hypervolume contributions. In: Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms (FOGA), pp. 103–112 (2009)Google Scholar
  5. 5.
    Bringmann, K., Friedrich, T.: The maximum hypervolume set yields near-optimal approximation. In: Pelikan, M., Branke, J. (eds.) GECCO 2010, pp. 511–518. ACM Press, New York (2010)Google Scholar
  6. 6.
    Corne, D., Knowles, J.D.: Some multiobjective optimizers are better than others. In: Proceedings of the 2003 Congress on Evolutionary Computation (CEC 2003), vol. 4, pp. 2506–2512. IEEE Press, Piscataway (2003)CrossRefGoogle Scholar
  7. 7.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 181–197 (2002)CrossRefGoogle Scholar
  8. 8.
    Fleischer, M.: The measure of Pareto optima. applications to multi-objective metaheuristics. In: Fonseca, C.M., et al. (eds.) EMO 2003. LNCS, vol. 2632, pp. 519–533. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Hanne, T.: On the convergence of multiobjective evolutionary algorithms. European Journal of Operational Research 117(3), 553–564 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hansen, M.P.: Metaheuristics for multiple objective combinatorial optimization. Ph.D. thesis, Institute of Mathematical Modelling, Technical University of Denmark (1998)Google Scholar
  11. 11.
    Knowles, J.D.: Local-Search and Hybrid Evolutionary Algorithms for Pareto Optimization. Ph.D. thesis, University of Reading, UK (2002)Google Scholar
  12. 12.
    Knowles, J.D., Corne, D.: On metrics for comparing non-dominated sets. In: Proceedings of the 2002 Congress on Evolutionary Computation Conference (CEC 2002), pp. 711–716. IEEE Press, Piscataway (2002)Google Scholar
  13. 13.
    Knowles, J.D., Corne, D.: Properties of an adaptive archiving algorithm for storing nondominated vectors. IEEE Transactions on Evolutionary Computation 7(2), 100–116 (2003)CrossRefGoogle Scholar
  14. 14.
    Knowles, J.D., Corne, D.: Bounded Pareto archiving: Theory and practice. In: Gandibleux, X., Sevaux, M., Sörensen, K., T’kindt, V. (eds.) Metaheuristics for Multiobjective Optimisation. Lecture Notes in Economics and Mathematical Systems, pp. 39–64. Springer, Berlin (2004)CrossRefGoogle Scholar
  15. 15.
    Laumanns, M.: Stochastic convergence of random search to fixed size Pareto set approximations. Arxiv preprint arXiv:0711.2949 (2007)Google Scholar
  16. 16.
    Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diversity in evolutionary multiobjective optimization. Evolutionary Computation 10(3), 263–282 (2002)CrossRefGoogle Scholar
  17. 17.
    Rudolph, G., Agapie, A.: Convergence properties of some multi-objective evolutionary algorithms. In: Proceedings of the 2000 Congress on Evolutionary Computation (CEC 2000), vol. 2, pp. 1010–1016. IEEE Press, Piscataway (2000)Google Scholar
  18. 18.
    Veldhuizen, D.A.V., Lamont, G.B.: Evolutionary computation and convergence to a Pareto front. In: Koza, J.R. (ed.) Late Breaking Papers at the Genetic Programming 1998 Conference, pp. 221–228. Stanford University Bookstore, Stanford University (1998)Google Scholar
  19. 19.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Giannakoglou, K., et al. (eds.) Evolutionary Methods for Design, Optimisation and Control, pp. 95–100. CIMNE, Barcelona (2002)Google Scholar
  20. 20.
    Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms - A comparative case study. In: Eiben, A.E., et al. (eds.) PPSN V 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  21. 21.
    Zitzler, E., Thiele, L., Bader, J.: SPAM: Set preference algorithm for multiobjective optimization. In: Rudolph, G., et al. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 847–858. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Grunert da Fonseca, V.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Manuel López-Ibáñez
    • 1
  • Joshua Knowles
    • 2
  • Marco Laumanns
    • 3
  1. 1.IRIDIA, CoDEUniversité Libre de BruxellesBrusselsBelgium
  2. 2.School of Computer ScienceUniversity of ManchesterUK
  3. 3.IBM ResearchZurichSwitzerland

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