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An Empirical Investigation of the Optimality and Monotonicity Properties of Multiobjective Archiving Methods

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Evolutionary Multi-Criterion Optimization (EMO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11411))

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Abstract

Most evolutionary multiobjective optimisation (EMO) algorithms explicitly or implicitly maintain an archive for an approximation of the Pareto front. A question arising is whether existing archiving methods are reliable with respect to their convergence and approximation ability. Despite theoretical results available, it remains unknown how these archivers actually perform in practice. In particular, what percentage of solutions in their final archive are Pareto optimal? How frequently do they experience deterioration during the archiving process? Deterioration means archiving a new solution which is dominated by some solution discarded previously. This paper answers the above questions through a systematic investigation of eight representative archivers on 37 test instances with two to five objectives. We have found that (1) deterioration happens to all the archivers; (2) the deterioration degree can vary dramatically on different problems; (3) some archivers clearly perform better than others; and (4) several popular archivers sometime return a population with most solutions being the non-optimal. All of these suggest the need of improvement of current archiving methods.

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Notes

  1. 1.

    For EMO algorithms without considering an external archive (e.g., NSGA-II [8]), their population can also be seen as an implicit archive where the selection operation is performed to preserve the best solutions ever produced [40].

  2. 2.

    The method of computing the dominated hypervolume in SMS-EMOA was from [13], available at http://iridia.ulb.ac.be/~manuel/hypervolume.

  3. 3.

    Here, “Pareto optimal” means being nondominated to all the solutions found during the run, rather than the problem’s Pareto optimal solutions.

References

  1. Aguirre, H., Tanaka, K.: Working principles, behavior, and performance of MOEAs on MNK-landscapes. Eur. J. Oper. Res. 181(3), 1670–1690 (2007)

    Article  Google Scholar 

  2. Aguirre, H., Zapotecas, S., Liefooghe, A., Verel, S., Tanaka, K.: Approaches for many-objective optimization: analysis and comparison on MNK-landscapes. In: Bonnevay, S., Legrand, P., Monmarché, N., Lutton, E., Schoenauer, M. (eds.) EA 2015. LNCS, vol. 9554, pp. 14–28. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-31471-6_2

    Chapter  Google Scholar 

  3. Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181(3), 1653–1669 (2007)

    Article  Google Scholar 

  4. Bezerra, L.C.T., López-Ibánez, M., Stützle, T.: Automatic component-wise design of multiobjective evolutionary algorithms. IEEE Trans. Evol. Comput. 20(3), 403–417 (2016)

    Article  Google Scholar 

  5. Bezerra, L.C.T., López-Ibáñez, M., Stützle, T.: A large-scale experimental evaluation of high-performing multi-and many-objective evolutionary algorithms. Evol. Comput. (2018, in press)

    Google Scholar 

  6. Corne, D., Knowles, J.: Some multiobjective optimizers are better than others. In: The 2003 Congress on Evolutionary Computation, vol. 4, pp. 2506–2512. IEEE (2003)

    Google Scholar 

  7. Deb, K., Jain, H.: An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part I: solving problems with box constraints. IEEE Trans. Evol. Comput. 18(4), 577–601 (2014)

    Article  Google Scholar 

  8. Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)

    Article  Google Scholar 

  9. 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, Berlin (2005). https://doi.org/10.1007/1-84628-137-7_6

    Chapter  MATH  Google Scholar 

  10. Fieldsend, J.E.: University staff teaching allocation: formulating and optimising a many-objective problem. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO), pp. 1097–1104. ACM (2017)

    Google Scholar 

  11. Fieldsend, J.E., Everson, R.M., Singh, S.: Using unconstrained elite archives for multiobjective optimization. IEEE Trans. Evol. Comput. 7(3), 305–323 (2003)

    Article  Google Scholar 

  12. Fonseca, C., Fleming, P.: An overview of evolutionary algorithms in multiobjective optimization. Evol. Comput. 3(1), 1–16 (1995)

    Article  Google Scholar 

  13. Fonseca, C.M., Paquete, L., López-Ibánez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: Proceedings of IEEE Congress Evolutionary Computation CEC 2006, pp. 1157–1163 (2006)

    Google Scholar 

  14. Hanne, T.: On the convergence of multiobjective evolutionary algorithms. Eur. J. Oper. Res. 117(3), 553–564 (1999)

    Article  MathSciNet  Google Scholar 

  15. Huband, S., Hingston, P., Barone, L., While, L.: A review of multiobjective test problems and a scalable test problem toolkit. IEEE Trans. Evol. Comput. 10(5), 477–506 (2006)

    Article  Google Scholar 

  16. Ishibuchi, H., Hitotsuyanagi, Y., Tsukamoto, N., Nojima, Y.: Many-objective test problems to visually examine the behavior of multiobjective evolution in a decision space. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN 2010. LNCS, vol. 6239, pp. 91–100. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15871-1_10

    Chapter  Google Scholar 

  17. Jain, H., Deb, K.: An evolutionary many-objective optimization algorithm using reference-point based nondominated sorting approach, part II: handling constraints and extending to an adaptive approach. IEEE Trans. Evol. Comput. 18(4), 602–622 (2014)

    Article  Google Scholar 

  18. Jin, H., Wong, M.-L.: Adaptive, convergent, and diversified archiving strategy for multiobjective evolutionary algorithms. Expert Syst. Appl. 37(12), 8462–8470 (2010)

    Article  Google Scholar 

  19. Judt, L., Mersmann, O., Naujoks, B.: Non-monotonicity of observed hypervolume in 1-Greedy S-Metric selection. J. Multi-Criteria Decis. Anal. 20(5–6), 277–290 (2013)

    Article  Google Scholar 

  20. Knowles, J., Corne, D.: Properties of an adaptive archiving algorithm for storing nondominated vectors. IEEE Trans. Evol. Comput. 7(2), 100–116 (2003)

    Article  Google Scholar 

  21. Knowles, J., Corne, D.: Bounded Pareto archiving: theory and practice. In: Gandibleux, X., Sevaux, M., Sörensen, K., T’kindt, V., et al. (eds.) LNE, vol. 535, pp. 39–64. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-642-17144-4_2

    Chapter  Google Scholar 

  22. Knowles, J.D., Corne, D.W., Fleischer, M.: Bounded archiving using the Lebesgue measure. In: The 2003 Congress on Evolutionary Computation, vol. 4, pp. 2490–2497. IEEE (2003)

    Google Scholar 

  23. Köppen, M., Yoshida, K.: Substitute distance assignments in NSGA-II for handling many-objective optimization problems. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 727–741. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-70928-2_55

    Chapter  Google Scholar 

  24. Kursawe, F.: A variant of evolution strategies for vector optimization. In: Schwefel, H.-P., Männer, R. (eds.) PPSN 1990. LNCS, vol. 496, pp. 193–197. Springer, Heidelberg (1991). https://doi.org/10.1007/BFb0029752

    Chapter  Google Scholar 

  25. Laumanns, M., Thiele, L., Deb, K., Zitzler, E.: Combining convergence and diversity in evolutionary multiobjective optimization. Evol. Comput. 10(3), 263–282 (2002)

    Article  Google Scholar 

  26. Laumanns, M., Zenklusen, R.: Stochastic convergence of random search methods to fixed size Pareto front approximations. Eur. J. Oper. Res. 213(2), 414–421 (2011)

    Article  MathSciNet  Google Scholar 

  27. Li, M., Grosan, C., Yang, S., Liu, X., Yao, X.: Multi-line distance minimization: a visualized many-objective test problem suite. IEEE Trans. Evol. Comput. 22(1), 61–78 (2018)

    Article  Google Scholar 

  28. Li, M., Yang, S., Liu, X.: Shift-based density estimation for Pareto-based algorithms in many-objective optimization. IEEE Trans. Evol. Comput. 18(3), 348–365 (2014)

    Article  Google Scholar 

  29. Li, M., Yang, S., Liu, X.: A test problem for visual investigation of high-dimensional multi-objective search. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC), pp. 2140–2147 (2014)

    Google Scholar 

  30. Li, M., Yang, S., Liu, X.: Pareto or non-pareto: bi-criterion evolution in multiobjective optimization. IEEE Trans. Evol. Comput. 20(5), 645–665 (2016)

    Article  Google Scholar 

  31. López-Ibáñez, M., Knowles, J., Laumanns, M.: On sequential online archiving of objective vectors. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds.) EMO 2011. LNCS, vol. 6576, pp. 46–60. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19893-9_4

    Chapter  Google Scholar 

  32. Reed, P.M., Hadka, D., Herman, J.D., Kasprzyk, J.R., Kollat, J.B.: Evolutionary multiobjective optimization in water resources: the past, present, and future. Adv. Water Resour. 51, 438–456 (2013)

    Article  Google Scholar 

  33. Rudolph, G., Agapie, A.: Convergence properties of some multi-objective evolutionary algorithms. In: Proceedings of the 2000 Congress on Evolutionary Computation, vol. 2, pp. 1010–1016. IEEE (2000)

    Google Scholar 

  34. Schaffer, J.D.: Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the First International Conference on Genetic Algorithms and their Applications, pp. 93–100 (1985)

    Google Scholar 

  35. Schütze, O., Laumanns, M., Coello, C.A., Dellnitz, M., Talbi, E.G.: Convergence of stochastic search algorithms to finite size Pareto set approximations. J. Glob. Optim. 41(4), 559–577 (2008)

    Article  MathSciNet  Google Scholar 

  36. Vlennet, R., Fonteix, C., Marc, I.: Multicriteria optimization using a genetic algorithm for determining a Pareto set. Int. J. Syst. Sci. 27(2), 255–260 (1996)

    Article  Google Scholar 

  37. Zhang, Q., Li, H.: MOEA/D: a multiobjective evolutionary algorithm based on decomposition. IEEE Trans. Evol. Comput. 11(6), 712–731 (2007)

    Article  Google Scholar 

  38. Zitzler, E., Deb, K., Thiele, L.: Comparison of multiobjective evolutionary algorithms: empirical results. Evol. Comput. 8(2), 173–195 (2000)

    Article  Google Scholar 

  39. Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Yao, X., et al. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30217-9_84

    Chapter  Google Scholar 

  40. Zitzler, E., Laumanns, M., Bleuler, S.: A tutorial on evolutionary multiobjective optimization. In: Gandibleux, X., Sevaux, M., Sörensen, K., T’kindt, V., et al. (eds.) Metaheuristics for Multiobjective Optimisation. LNE, vol. 535, pp. 3–37. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-642-17144-4_1

    Chapter  MATH  Google Scholar 

  41. Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Evolutionary Methods for Design, Optimisation and Control, pp. 95–100, Barcelona, Spain (2002)

    Google Scholar 

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Acknowledgement

This work was supported by EPSRC (Grant Nos. EP/J017515/1 and EP/P005578/1) and Science and Technology Innovation Committee Foundation of Shenzhen (Grant Nos. ZDSYS201703031748284 and JCYJ20170307105521943).

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Authors and Affiliations

  1. CERCIA, School of Computer Science, University of Birmingham, Birmingham, B15 2TT, UK

    Miqing Li & Xin Yao

  2. Shenzhen Key Laboratory of Computational Intelligence (SKyLoCI), Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen, People’s Republic of China

    Xin Yao

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  1. Miqing Li
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Correspondence to Miqing Li .

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Editors and Affiliations

  1. Michigan State University, East Lansing, MI, USA

    Kalyanmoy Deb

  2. Michigan State University, East Lansing, MI, USA

    Erik Goodman

  3. CINVESTAV-IPN, Mexico, Mexico

    Carlos A. Coello Coello

  4. University of Wuppertal, Wuppertal, Germany

    Kathrin Klamroth

  5. University of Jyvaskyla, Jyväskylä, Finland

    Kaisa Miettinen

  6. Otto von Guericke University Magdeburg, Magdeburg, Germany

    Sanaz Mostaghim

  7. Cornell University, Ithaca, NY, USA

    Patrick Reed

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Li, M., Yao, X. (2019). An Empirical Investigation of the Optimality and Monotonicity Properties of Multiobjective Archiving Methods. In: Deb, K., et al. Evolutionary Multi-Criterion Optimization. EMO 2019. Lecture Notes in Computer Science(), vol 11411. Springer, Cham. https://doi.org/10.1007/978-3-030-12598-1_2

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  • DOI: https://doi.org/10.1007/978-3-030-12598-1_2

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